11institutetext: Department of Applied Mathematics, University of Waterloo,
Waterloo, Ontario N2L 3G1, Canada22institutetext: Perimeter Institute for
Theoretical Physics,
31 Caroline St. N.,Waterloo, Ontario N2L 2Y5, Canada33institutetext: School of
Mathematical and Physical Sciences, Macquarie University,
Sydney, NSW 2109, Australia

# Exotic Black Hole Thermodynamics in Third-Order Lovelock Gravity

Brayden Hull 3 Fil Simovic<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The generalization of Birkhoff’s theorem to higher dimensions in Lovelock
gravity allows for black hole solutions with horizon geometries of non-
constant curvature. We investigate thermodynamic aspects of these ‘exotic’
black hole solutions, with a particular emphasis on their phase transitions.
We consider an extended phase space where the cosmological constant acts as a
thermodynamic pressure, and examine both uncharged and $U(1)$ charged
solutions. In $d=7$, black hole solutions are restricted to having constant
curvature horizon base manifolds. Uncharged $d=7$ black holes possess novel
triple point phenomena analogous to those recently uncovered in exotic $d=6$
black holes in Gauss-Bonnet gravity, while their charged counterparts
generically undergo small-large black hole phase transitions. In $d=8$, we
find that both charged and uncharged black holes exhibit triple point
behaviour and small-large black hole transitions. We also show that a wide
range of ‘exotic’ horizon geometries can be ruled out due to the appearance of
naked singularities.

###### Keywords:

black hole thermodynamics, Lovelock gravity, phase transitions

##  1 Introduction

Reconciling the distinctly quantum mechanical nature of Standard Model field
theory with the classical notion of gravity as being emergent from the
geometry of semi-Riemannian manifolds remains one of the most important open
problems in theoretical physics, and while decades of active research have
yielded significant progress, much remains to be understood carlip2015 . One
of the most promising paths towards understanding the would-be quantum
features of the gravitational interaction has been the study of its
thermodynamic properties. The extent to which a thermodynamic description of
the gravitational field can be formulated (and understood to arise from its
microscopic degrees of freedom) has been explored extensively over the years,
both borrowing from and contributing to lessons learned in string theory,
quantum field theory in curved space, and information theory. The genesis of
the subject can be traced back to the early 1970’s when Bardeen, Carter, and
Hawking, along with observations by Bekenstein, showed that equilibrium black
hole configurations in asymptotically flat spacetime obey a law closely
resembling the first law of thermodynamics of ordinary systems bardeen1973a ;
bekenstein1973a . The fact that black holes (which are classically understood
to be purely geometric objects) obey such a law has guided research into the
thermodynamic properties of black holes ever since wald2000 ; carlip2014 .

Somewhat dramatically, black holes have been shown to undergo phase
transitions akin to those seen in ordinary thermodynamic systems. The
prototypical example is the Hawking-Page transition between an anti-de Sitter
black hole and thermal radiation hawking1983a . Distinct from the evaporation
process, this transition has an important interpretation as a deconfinement
transition in gauge theory degrees of freedom through the anti-de
Sitter/conformal field theory correspondence, and has provided significant
insight into the structure of strongly coupled field theories, the QCD phase
diagram, and even unitarity as it relates to the black hole information
paradox maldacena1998 ; cui2021 ; harlow2016 . The likeness between black hole
thermodynamics and that of traditional matter systems becomes even more
salient in the extended phase space (where the cosmological constant is
treated as a thermodynamic pressure—see kastor2009 ) where not only do more
exotic transitions appear, but also the notion of holographic heat engines and
many other interesting phenomena that have been extensively explored over the
past few years (see kubiznak2011  for a review).

The richest phenomenology is observed in asymptotically anti-de Sitter (AdS)
black hole solutions in Lovelock gravity, a higher-dimensional generalization
of Einstein gravity which is ordinarily constructed in $d>4$111Recent attempts
at formulating Gauss-Bonnet gravity in $d=4$ indicate it may be possible to
obtain nontrivial deviations from general relativity even in this case
glavan2020 ; hennigar2020a . lovelock1971 . Such black holes have been shown
to exhibit not only the ordinary Hawking-Page transition, but also reentrant
phase transitions, triple points, superfluid-like transitions, and more
kubiznak2012 ; altamirano2013 ; frassino2014 ; hennigar2017f . The theory
defined by the Lovelock Lagrangian is the most general metric theory of
gravity with second-order equations of motion (being free of ghost
instabilities), and differs from general relativity through the inclusion of
higher-curvature terms to the Einstein-Hilbert action. Lovelock gravity has
enjoyed significant theoretical interest, as these corrections to Einstein
gravity arise naturally in quantum gravity—for example, the Gauss-Bonnet term
appears as the leading correction in 10d gauged supergravity and as a quantum
correction from heterotic string theory cecotti1988 .

Lovelock gravity possesses another interesting feature that remains largely
unexplored in the context of black hole thermodynamics: the evasion of
Birkhoff’s theorem for general relativity. Ordinarily the well-known
Schwarzschild solution

$ds^{2}=-f(r,t)dt^{2}+\frac{dr^{2}}{f(r,t)}+r^{2}d\Sigma_{d-2}^{2}$ (1)

is unique—although the base manifold $d\Sigma_{d-2}^{2}$ is in principle an
arbitrarily chosen Einstein manifold, the $S^{2}$ is the only possibility in
four dimensions. This theorem has a natural generalization to higher
dimensions in terms of the Schwarzschild-Tangherlini solution tangherlini1963
. Birkhoff’s theorem also holds in Lovelock gravity deser2005 ; zegers2005
where unlike $d=4$ general relativity, there is enough freedom to have
arbitrary base manifolds with both constant and non-constant curvature222In
fact, this is a generic feature of higher-dimensional metric theories of
gravity whose Lagrangians consist of invariants constructed from the Riemann
tensor and its covariant derivatives MasatoSeiju ; hervik2020 . dotti2005 ;
ray2015 ; Dotti_2007 ; DOTTI_2009 ; Dotti_2010 ; Oliva_2013 ; sourya2015 . In
this setting, the constants characterizing the Euler invariants and Lovelock
tensors appear in the metric function, and determine the transverse part of
the metric. A large new class of black hole solutions therefore exist in
Lovelock gravity with ‘warped’ base manifold geometries. The implications of
such modifications for the thermodynamic features of black hole solutions are
relatively unknown, with the case of Gauss-Bonnet gravity being only very
recently examined hull2021 . In this work, we aim to extend this analysis to
third-order Lovelock gravity, where there is significantly more freedom in the
choice of horizon geometry for these exotic black holes.

This paper is organized as follows. In Section 2, we discuss the formulation
of Lovelock gravity, and introduce the ‘exotic’ black hole solutions we will
study herein. In Section 3, we define the relevant thermodynamic quantities
associated with general black hole solutions in Lovelock gravity. In Section
4, we introduce the solutions considered throughout and derive the relevant
thermodynamic quantities. In Section 5, we study third-order Lovelock black
holes in $d=7$, focusing on both the charged and uncharged solutions. We
examine the phase structure through the free energy, and study the equation of
state. In Section 6, we turn our attention to $d=8$ solutions, the lowest
dimension where non-constant curvature base manifolds are allowed. We again
examine the phase structure and equation of state, and detail the impact that
deviations from constant curvature have on the thermodynamic properties of the
solutions. We close in Section 7 with a summary of the main findings and
avenues for future investigations. Except for Sections 2 and 3, we use natural
units where $c=\hbar=G_{N}=1$.

##  2 Lovelock Gravity and Exotic Black Holes

In this section, we briefly review Lovelock theory and discuss how black hole
solutions with exotic horizon geometries arise.

###  2.1 Lovelock Gravity

Lovelock gravity represents a large class of classical metric gravitational
theories that all possess a number of desirable properties, namely, having
Lagrangians that are manifestly diffeomorphism-invariant and lead to second-
order equations of motion333This means in particular that there are no linear
instabilities in any momentum degrees of freedom in the corresponding
Hamiltonian (the Ostragradsky instability).. Such theories differ from general
relativity (which is the unique Lovelock theory in four dimensions444Attempts
to formulate Lovelock gravity in $d=4$ have been made to various degrees of
success, see e.g. glavan2020 ; hennigar2020a .) through the inclusion of
higher-curvature terms in the Einstein-Hilbert action, which ordinarily
contribute to the equations of motion only in higher dimensions (when $d>4$).
The Lovelock Lagrangian is

$\mathcal{L}=\frac{1}{16\pi G}\sum_{k=0}^{K}\hat{\alpha}_{k}\mathcal{L}^{(k)}\
,$ (2)

where $\hat{\alpha}_{k}$ are the Lovelock coupling constants and
$\mathcal{L}^{(k)}$ are the Euler densities, which are constructed through
various contractions of the Riemann tensor $R_{abcd}$ as

$\mathcal{L}^{(k)}=\frac{1}{2^{k}}\,\delta_{c_{1}d_{1}\ldots
c_{k}d_{k}}^{a_{1}b_{1}\ldots
a_{k}b_{k}}\,{R_{a_{1}b_{1}}}^{\\!c_{1}d_{1}}\ldots{R_{a_{k}b_{k}}}^{\\!c_{k}d_{k}}\
.$ (3)

With $\mathcal{L}^{(0)}=1$, $\hat{\alpha}_{0}=-2\Lambda$,
$\mathcal{L}^{(1)}=R$ and $\hat{a}_{1}=1$, the $K=1$ truncation of (2) yields
the Einstein-Hilbert action. Notably, the $k$-th Euler density is a
topological invariant when $d\leq 2k$, and therefore does not contribute to
the equations of motion of the resulting theory. Nontrivial deviations from
general relativity therefore occur only when the dimension of spacetime is
large enough for a given finite truncation of the sum (2), namely when $d>2K$.

The full $d$-dimensional action of Lovelock theory with matter can thus be
written as

$S=\int d^{d}x\sqrt{-g}\left(\frac{1}{16\pi
G_{N}}\sum_{k=0}^{K}\hat{\alpha}_{k}\mathcal{L}^{(k)}+\mathcal{L}_{\text{matter}}\right)\
,$ (4)

whose variation with respect to the metric yields the equations of motion

$\sum_{k}^{K}\hat{\alpha}_{k}\mathcal{G}_{ab}^{(k)}=8\pi G_{N}T_{ab}\ ,$ (5)

where $T_{ab}$ is the stress-energy tensor and $\mathcal{G}^{(k)}_{ab}$ are
the Lovelock tensors, given by

$\mathcal{G}_{ab}^{(k)}=-\frac{1}{2^{(k+1)}}\,g_{za}\,\delta_{be_{1}f_{1}\ldots
e_{k}f_{k}}^{zc_{1}d_{1}\ldots
c_{k}d_{k}}\,{R_{c_{1}d_{1}}}^{e_{1}f_{1}}\ldots{R_{c_{k}d_{k}}}^{e_{k}f_{k}}\
.$ (6)

We will be interested in both uncharged and $U(1)$ charged black holes, the
latter involving an additional electromagnetic term in the action:
$\mathcal{L}_{\text{matter}}=-4\pi G_{N}F_{ab}F^{ab}$. The resulting field
equations are then

$\sum_{k=0}^{K}\hat{\alpha}_{(k)}\mathcal{G}_{ab}^{(k)}=8\pi
G_{N}\left(F_{ac}F_{b}^{c}-\frac{1}{4}g_{ab}F_{cd}F^{cd}\right).$ (7)

We will use the following spherically symmetric ansatz for the metric and
gauge field:

$ds^{2}={\textsf{g}}_{ij}{dy^{i}dy^{j}}+{\gamma}_{\alpha\beta}dx^{\alpha}dx^{\beta}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Sigma_{d-2}^{2}\
,\quad F=\frac{Q}{r^{d-2}}dt\wedge dr\ .$ (8)

Latin indices are reserved for coordinates in the $t-r$ plane while Greek
indices refer to coordinates on the base manifold $d\Sigma_{d-2}^{2}$. The
importance of this metric is how one defines $\gamma_{\alpha\beta}$, which
determines the geometry of the transverse sections (including the horizon). In
this work we take the base manifold with metric $\gamma_{\alpha\beta}$ to be a
$(d-2)$ compact space not necessarily of constant curvature. The constant
curvature case (corresponding to ‘ordinary’ Lovelock black holes) has been
studied extensively frassino2014 , while the non-constant curvature case has
only been examined in the context of Gauss-Bonnet gravity hull2021 . Inserting
the metric (8) into the field equations produces two simplified expressions:

$\begin{split}\mathcal{G}_{j}^{i}&\equiv\frac{-(d-2)\delta_{j}^{i}}{2r^{d-2}}\frac{d}{dr}\sum_{n=0}^{K}\left\\{b_{n}\left(r^{d-2n-1}A_{n}\left(\frac{-f(r)}{r^{2}}\right)\right)\right\\}=8\pi
G_{N}T_{j}^{i}\ ,\\\
\mathcal{G}^{\alpha}_{\beta}&\equiv\frac{-\delta^{\alpha}_{\beta}}{2r^{d-3}}\frac{d^{2}}{dr^{2}}\sum_{n=0}^{K}\left\\{b_{n}\left(r^{d-2n-1}A_{n}\left(\frac{-f(r)}{r^{2}}\right)\right)\right\\}=8\pi
G_{N}T^{\alpha}_{\beta}\ ,\end{split}$ (9)

where

$A_{n}\left(\frac{-f(r)}{r^{2}}\right)\equiv\sum_{k=n}^{K}\alpha_{k}\binom{k}{n}\left(\frac{-f(r)}{r^{2}}\right)^{k-n}\
,$ (10)

and the $b_{n}$ are arbitrary constants which depend on the geometry of the
base manifold $d\Sigma_{d-2}^{2}$ (see Section 2.2). These field equations
determine the metric function $f(r)$, which is obtained as a solution to the
following polynomial of degree $K$ in $f(r)$:

$\sum_{n=0}^{K}\frac{b_{n}}{r^{2n}}\left(\sum_{k=n}^{K}\alpha_{k}\binom{k}{n}\left(\frac{-f(r)}{r^{2}}\right)^{k-n}\right)=\frac{16\pi
G_{N}M}{(d-2)\Sigma_{d-2}r^{d-1}}-\frac{8\pi
G_{N}Q^{2}}{(d-2)(d-3)}\frac{1}{r^{2d-4}}$ (11)

Here $M$ represents the ADM mass, and $Q$ is the charge defined by

$Q=\frac{1}{2\Sigma_{d-2}}\int\star\,F\ .$ (12)

Finally, $\alpha_{k}$ are the re-scaled Lovelock coupling constants defined by

$\alpha_{0}=\frac{\hat{\alpha}_{(0)}}{(d-1)(d-2)}=\quad\alpha_{1}=\hat{\alpha}_{(1)},\quad\alpha_{k}=\hat{\alpha}_{(k)}\prod_{n=3}^{2k}(d-n)\quad\text{
for }\quad k\geq 2\ .$ (13)

Note that constant curvature base manifolds have $b_{n}=\kappa^{n}$ with
$\kappa\in\\{-1,0,+1\\}$, corresponding to negative, flat and positive
curvature respectively. This can be shown by direct comparison of (11) with
the familiar form of the constant curvature $f(r)$ polynomial given by

$\sum_{k=0}^{K}\alpha_{k}\left(\frac{\kappa-f}{r^{2}}\right)^{k}=\frac{16\pi
G_{N}M}{(d-2)\Sigma_{d-2}^{(\kappa)}r^{d-1}}-\frac{8\pi
G_{N}Q^{2}}{(d-2)(d-3)}\frac{1}{r^{2d-4}}\ .$ (14)

where $M$, $Q$ and the coupling constants are as defined above, and
$\Sigma_{d-2}^{(\kappa)}$ is the volume of the constant curvature compact
space.

In what follows we take $K=3$ in the above expressions, corresponding to
third-order Lovelock gravity, and set $\alpha_{1}=1$ in order to recover
general relativity in the $\\{\alpha_{2},\alpha_{3}\\}\rightarrow 0$ limit.
Useful for analyzing the structure of the resulting solutions is the
Kretschmann scalar, a curvature invariant which for our general metric (8) can
be written as

$\displaystyle\mathcal{K}$ $\displaystyle=R^{abcd}R_{abcd}$ (15)
$\displaystyle=\left(\frac{d^{2}f(r)}{dr^{2}}\right)^{2}+\frac{2(d-2)}{r^{2}}\left(\frac{df(r)}{dr}\right)^{2}+\frac{2(d-2)(d-3)f(r)^{2}}{r^{4}}-\frac{4R[\gamma]f(r)}{r^{4}}+\frac{\mathcal{K}[\gamma]}{r^{4}}\
.$

It is a function of both the metric function $f(r)$ and the Ricci and
Kretschmann scalars of the base manifold ($R[\gamma]$ and
$\mathcal{K}[\gamma]$, respectively), and provides a coordinate-free measure
of curvature which can be used to locate singularities and other pathologies
in the spacetime.

###  2.2 Exotic Horizon Geometries

The primary motivation of this work is to examine the role that horizon
geometry plays in determining the thermodynamic features of black hole
spacetimes. Ordinarily, Birkhoff’s theorem in general relativity strictly
limits the types of allowed horizon geometries. For a general spherically
symmetric metric of the form

$ds^{2}=-f(r,t)dt^{2}+\frac{dr^{2}}{f(r,t)}+r^{2}d\Sigma_{d-2}^{2}\ ,$ (16)

the transverse part of the metric $d\Sigma_{d-2}^{2}$ must be an Einstein
manifold (one for which $R_{\mu\nu}=kg_{\mu\nu}$ in local coordinates for some
constant $k$) with spherical symmetry555Absent spherical symmetry one may also
consider $\mathcal{H}^{2}$, $\text{I\\!R}^{2}$, $T^{2}$, $\text{I\\!R}P^{2}$,
and others.. In $d=4$ there is only one possibility, namely $\mathcal{S}^{2}$,
implying that all vacuum solutions with $SO(3)$ symmetry and positive
curvature are diffeomorphic to the maximal extension of the Schwarzschild
solution in an open subset. In higher dimensions other Einstein manifolds are
allowed, such as the Bohm metrics [ref].

In Lovelock gravity the base manifold $d\Sigma_{d-2}^{2}$ need not be an
Einstein manifold, allowing for a rich variety of horizon geometries ray2015 .
In fact, $d\Sigma_{d-2}^{2}$ can be rather arbitrary, the only requirement
being that various constraints are satisfied which relate the Lovelock tensors
of the base manifold. Specifically, the metric function $f(r)$ is a solution
to the polynomial (11) with the additional requirement that

$\hat{\mathcal{G}}_{\beta}^{(n)\alpha}=-\frac{(d-3)!\,b_{n}}{2(d-2n-3)!}\delta_{\beta}^{\alpha}\
,$ (17)

where $\hat{\mathcal{G}}_{\beta}^{(n)\alpha}$ are simply the Lovelock tensors
of the base manifold. The constants $b_{n}$ appear in the polynomial (11)
defining the metric function and are arbitrary, though they depend on the
geometry of the base manifold through (17).

When $d=2K+1$ the tranverse space must be of constant curvature Dotti_2007 ;
farhangkhah2014 , so the first dimension in which the $K$-th Lovelock term
begins to contribute only admits spherical, flat, and hyperbolic transverse
sections and has $b_{n}=\kappa^{n}$ with $\kappa\in\\{-1,0,+1\\}$. For third-
order Lovelock gravity, this means that only black hole solutions in $d\geq 8$
can possess exotic horizon geometries. Furthermore, we emphasize that
significant restrictions on the allowed geometries arise from the field
equations. For example, the Bohm metrics

$ds^{2}=d\rho^{2}+a(\rho)^{2}d\Omega_{p}^{2}+b(\rho)^{2}d\Omega_{q}^{2}$ (18)

represent an infinite family of local metrics that can be extended to
manifolds of topology $S^{p+q+1}$. Black holes with Bohm horizons have non-
constant curvature, yet of the infinitely many metrics given by different
choices of $a(\rho)$ and $b(\rho)$ in (18), only one choice turns out to be
admissible as a horizon due to the Gauss-Bonnet correction appearing when
$K\geq 2$ dotti2005 . Higher orders in Lovelock theory likely further
constrain the allowed choices of horizon geometry, though this remains a
currently unexplored feature of exotic black holes in higher-dimensional
Lovelock gravity. In this work we focus on choices of $b_{n}$ that produce
everywhere real and analytic metric functions, ensuring that curvature scalars
are finite and that the spacetime has the appropriate asymptotics, rather than
attempting to derive general constraints on the $b_{n}$s. We will see that
many interesting features appear even in the limited parameter space we study
herein.

##  3 Extended Phase Space Thermodynamics

We operate in an extended thermodynamic phase space, which arises from
considering variations of the cosmological constant $\Lambda$ when
constructing the first law for black hole systems. The inclusion of such a
variation is motivated by the observation that the ‘ordinary’ first law of
black hole mechanics contains no pressure-volume term, since there are no
obvious notions of such quantities for a black hole. Since the cosmological
constant is equivalent to a perfect fluid stress-energy tensor, a variable
$\Lambda$ provides a mechanism for would-be pressure variations in a
gravitational system. Precisely, we identify the cosmological constant with a
fluid pressure through the relation

$P=-\frac{\Lambda}{8\pi}=-\frac{(d-1)(d-2)}{16\pi l^{2}}\ ,$ (19)

where $l$ is the AdS length scale. The consequences of such a variation were
first explored by Kastor and collaborators kastor2009 ; kastor2010a ;
kastor2011b , who showed through a Hamiltonian analysis that the resulting
first law is supplemented by a term $\Theta\delta\Lambda$, where the potential
$\Theta$ conjugate to $\Lambda$ plays the role of the thermodynamic volume,
and, at least for Schwarzschild-AdS black holes, is exactly the renormalized
geometric volume of AdS minus the volume of the black hole. In this way, a
pressure-volume term naturally enters into the first law. From this extended
phase space emerged the area of research known as black hole chemistry, so
named because the inclusion of pressure in the thermodynamic description gives
rise to a number of phenomena in the gravitational sector that bear striking
resemblance to thermodynamic phenomena observed in everyday systems such as
water, helium, and dry ice hennigar2017f ; kubiznak2017b . The first law in
the extended phase space formally reads

$\delta M=T\delta S+V\delta P\ ,$ (20)

where additional terms enter if angular momentum or charge are present. In
Lovelock theory, the mass $M$, temperature $T$, entropy $S$, and charge $Q$ of
static, asymptotically AdS black holes are given by hull2021

$\displaystyle\quad M$ $\displaystyle=\frac{\Sigma_{d-2}(d-2)}{16\pi
G_{N}}\sum_{k=0}^{K}\alpha_{k}b_{k}r_{+}^{d-1-2k}+\frac{\Sigma_{d-2}}{2(d-3)}\frac{Q^{2}}{r_{+}^{d-3}}\
,$ (21) $\displaystyle S$
$\displaystyle=\frac{\Sigma_{d-2}(d-2)}{4G_{N}}\sum_{k=0}^{K}\frac{kb_{k-1}\alpha_{k}r_{+}^{d-2k}}{d-2k}\
,$ (22) $\displaystyle T$ $\displaystyle=\frac{1}{4\pi
r_{+}D\left(r_{+}\right)}\left[\sum_{k=0}^{K}b_{k}\alpha_{k}(d-2k-1)r_{+}^{-2(k-1)}-\frac{8\pi
G_{N}Q^{2}}{(d-2)r_{+}^{2(d-3)}}\right]\ ,$ (23)

where $r_{+}$ is the largest positive real root of $f(r_{+})=0$ and we have
defined

$D\left(r_{+}\right)\equiv\sum_{k=1}^{K}k\alpha_{k}b_{k-1}r_{+}^{-2(k-1)}\ .$
(24)

These quantities obey the following first law and Smarr relation:

$\begin{split}\delta M&=T\delta S-\frac{1}{16\pi
G_{N}}\sum_{h}\hat{\Psi}^{(k)}\delta\hat{\alpha}_{(k)}+\Phi\delta Q\ ,\\\
(d-3)M&=(d-2)TS+\sum_{k}2(k-1)\frac{\hat{\Psi}^{(k)}\hat{\alpha}_{(k)}}{16\pi
G_{N}}+(d-3)\Phi Q\ .\end{split}$ (25)

Here, $\hat{\Psi}^{(k)}$ are potentials conjugate to the Lovelock coupling
constants $\hat{\alpha}_{(k)}$ which are promoted to thermodynamic variables.
The potentials are defined by

$\hat{\Psi}^{(k)}=4\pi T\mathcal{A}^{(k)}+\mathcal{B}^{(k)}+\Theta^{(k)}\ ,$
(26)

with

$\displaystyle\mathcal{B}^{(k)}$ $\displaystyle=-\frac{16\pi
kG_{N}M(d-1)!}{b(d-2k-1)!}\left(-\frac{1}{\ell^{2}}\right)^{k-1},\quad
b=\sum_{k}\frac{\hat{\alpha}_{k}k(d-1)!}{(d-2k-1)!}\left(-\frac{1}{\ell^{2}}\right)^{k-1}\
,$ (27) $\displaystyle\Theta^{(k)}$
$\displaystyle=\int_{\Sigma}\sqrt{-g}\mathcal{L}^{(k)}[s]-\int_{\Sigma_{\mathrm{AdS}}}\sqrt{-g_{\mathrm{AdS}}}\mathcal{L}^{(k)}\left[s_{\mathrm{AdS}}\right]\
.$

The thermodynamic entropy is also modified in a non-trivial way compared to
the Einstein-AdS case, and is no longer one-quarter the area of the horizon.
It is now given by

$S=\frac{1}{4G_{N}}\sum_{k}\hat{\alpha}_{k}\mathcal{A}^{(k)},\quad\mathcal{A}^{(k)}=k\int_{\mathcal{H}}\sqrt{\sigma}\mathcal{L}^{(k-1)}\
,$ (28)

where $\sigma$ is the induced metric on the black holes horizon and
$\mathcal{L}^{(k-1)}$ are the Euler densities of the horizon.

##  4 Exotic Black Holes in Third-Order Lovelock Gravity

In this section, we present general solutions for charged black hole metrics
in third-order Lovelock gravity with exotic horizon geometries. We solve the
polynomial (11) to determine the metric function, and examine various
constraints on the couplings $\\{\alpha_{i}\\}$ that arise from considerations
of analyticity and asymptotic behaviour. We also determine the equation of
state and free energy, and place bounds on the allowed pressure (cosmological
constant) for these black holes.

###  4.1 Solutions

Setting $K=3$ in (11) gives the following polynomial for the metric function
$f$:

$\displaystyle-\frac{\alpha_{3}f^{3}}{r^{6}}+\left(\frac{\alpha_{2}}{r^{4}}+\frac{3b_{1}\alpha_{3}}{r^{6}}\right)f^{2}$
$\displaystyle+\left(-\frac{1}{r^{2}}-\frac{2b_{1}\alpha_{2}}{r^{4}}-\frac{3b_{2}\alpha_{3}}{r^{6}}\right)f+\alpha_{0}+\frac{b_{1}}{r^{2}}+$
$\displaystyle\frac{b_{2}\alpha_{2}}{r^{4}}+\frac{b_{3}\alpha_{3}}{r^{6}}=A(r)\
,$ $\displaystyle A(r)$ $\displaystyle\equiv\frac{16\pi
M}{(d-2)\Sigma_{d-2}r^{d-1}}-\frac{8\pi Q^{2}}{(d-2)(d-3)r^{2d-4}}\ .$ (29)

This is a cubic equation for $f$ possessing three distinct solutions. The
first is given by

$f_{E}=\frac{\Big{(}r^{4}\left(3\alpha_{3}+\alpha_{2}^{2}\right)-9\left(b_{1}^{2}-b_{2}\right){\alpha_{3}}^{2}\Big{)}+\left(r^{2}\alpha_{2}+3\alpha_{3}b_{1}\right)\left(\tfrac{1}{2}\big{(}Y+\sqrt{X}\big{)}\right)^{1/3}-\left(\tfrac{1}{2}\big{(}Y+\sqrt{X}\big{)}\right)^{2/3}}{3\alpha_{3}\left({\tfrac{1}{4}\big{(}Y+\sqrt{X}}\big{)}\right)^{1/3}}$
(30)

where we have defined

$\displaystyle X\equiv$ $\displaystyle\
\Big{(}27\alpha_{3}^{3}\big{(}2b_{1}^{3}-3b_{1}b_{2}+b_{3}\big{)}-\big{(}27\alpha_{3}^{2}\big{(}A(r)-\alpha_{0}\big{)}-2\alpha_{2}^{3}+9\alpha_{2}\alpha_{3}\big{)}r^{6}\Big{)}^{2}$
$\displaystyle\
+4\Big{(}9\alpha_{3}^{2}\big{(}b_{2}-b_{1}^{2}\big{)}-\big{(}\alpha_{2}^{2}+3\alpha_{3}\big{)}r^{4}\Big{)}^{3}\
,$ $\displaystyle Y\equiv$ $\displaystyle\
(81b_{1}b_{2}-54b_{1}^{3}-27b_{3})\alpha_{3}^{3}+\Big{(}27\alpha_{3}^{2}\big{(}A(r)-\alpha_{0}\big{)}+9\alpha_{2}\alpha_{3}-2\alpha_{2}^{3}\Big{)}r^{6}\
.$

The subscript $E$ indicates that this branch gives the appropriate Einstein
limit when successively taking the limits $\alpha_{3}\rightarrow 0$ and
$\alpha_{2}\rightarrow 0$. The other two (conjugate) branches are given by the
solutions

$f=\frac{\left(1\mp
i\sqrt{3}\right)\Big{(}r^{4}\\!\left(3\alpha_{3}+\alpha_{2}^{2}\right)-9\left(b_{1}^{2}-b_{2}\right)\alpha_{3}^{2}\Big{)}+\left(r^{2}\alpha_{2}+3\alpha_{3}b_{1}\right)\\!\sqrt[3]{4\big{(}Y+\sqrt{X}\big{)}}+\left(1\pm
i\sqrt{3}\right)\left(\tfrac{1}{2}\big{(}Y+\sqrt{X}\big{)}\right)^{2/3}}{3\alpha_{3}\sqrt[3]{4\big{(}Y+\sqrt{X}}\big{)}}.$

These two solutions are distinguished by a lack of convergent
$\alpha_{2}\rightarrow 0$ limit. We further differentiate between the two by
considering their Gauss-Bonnet ($\alpha_{3}\rightarrow 0$) limits. The
solution $f_{GB}$, given by the upper sign, converges in this limit and yields
the appropriate metric function for the Schwarzschild-AdS black hole (when
$\\{b_{1},b_{2},b_{3}\\}=\\{\kappa,\kappa^{2},\kappa^{3}\\}$) and is therefore
referred to as the Gauss-Bonnet branch. The solution $f_{L}$, given by the
lower sign, has a divergent $\alpha_{3}\rightarrow 0$ limit and exists only in
third-order Lovelock gravity. We refer to this as the Lovelock branch.

A number of considerations arise from the form of the solutions (30)–(4.1). As
discussed in frassino2014 , a bound on the couplings
$\\{\alpha_{2},\alpha_{3}\\}$ can be obtained by demanding that all three
solutions to (4.1) are simultaneously real-valued. This is determined by
positivity of the discriminant of the cubic equation for $f$, which implies
the following constraint on the couplings:

$27{\alpha_{0}}^{2}{\alpha_{3}}^{2}+4\alpha_{0}\alpha_{2}^{3}-18\alpha_{0}\alpha_{2}\alpha_{3}-{\alpha_{2}}^{2}+4\alpha_{3}\leq
0$ (31)

When (31) is satisfied, one can smoothly take the Einstein limit by
appropriate choice of solution. In Figure 1, we plot regions in the parameter
space for which this condition is satisfied.

Figure 1: The discriminant condition for solutions to (4.1). Left: The shaded
area indicates regions of the parameter space where all solutions are real, as
determined by the constraint (31). Right: A fixed-$\alpha_{3}$ slice taken
from the plot on the left, showing the emergence of a minimum/maximum allowed
pressure for certain values of $\alpha_{2}$.

The inequality (31) puts an upper (and sometimes lower) bound on the allowed
pressure for certain values of $\alpha_{2}$, given in the Einstein and Gauss-
Bonnet cases by

$P_{\pm}=\frac{(d-1)(d-2)}{432\pi}\frac{9\alpha_{2}\alpha_{3}-2\alpha_{2}^{3}\pm
2\left(\alpha_{2}^{2}-3\alpha_{3}\right)^{\frac{3}{2}}}{\alpha_{3}^{2}}\ .$
(32)

Now introducing the following dimensionless variables

$\alpha=\frac{\alpha_{2}}{\sqrt{\alpha_{3}}}\ ,\quad p=4\sqrt{\alpha_{3}}P\
,r_{+}=v\alpha_{3}^{\frac{1}{4}},\quad
T=\frac{t\alpha_{3}^{-\frac{1}{4}}}{d-2},\quad m=\frac{16\pi
M}{(d-2)\Sigma_{d-2}^{(\kappa)}\alpha_{3}^{\frac{d-3}{4}}},\quad
Q=\frac{q}{\sqrt{2}}\alpha_{3}^{\frac{d-3}{4}}$ (33)

the pressure can be expressed as

$p_{\pm}=\frac{(d-1)(d-2)}{108\pi}\left(9\alpha-2\alpha^{3}\pm
2\left(\alpha^{2}-3\right)^{\frac{3}{2}}\right)\ .$ (34)

This matches our expectations for the constant curvature case, since these
bounds are sensitive only to the asymptotic structure of the solutions that
are independent of the horizon geometry. We see that both branches terminate
at the same values of $\alpha=\sqrt{3}$, below which the pressure becomes
imaginary and the spacetime is no longer asymptotically anti-de Sitter. We
therefore limit our analysis to couplings for which $\alpha\geq\sqrt{3}$.
There is also a narrow region in the parameter space where a minimum pressure
exists along with a maximum; in this case when $\sqrt{3}<\alpha<2$. When
$\alpha\geq 2$ there is no lower bound on the pressure.

###  4.2 Equation of State and Free Energy

For ordinary thermodynamic systems, the equation of state provides a closure
relation between thermodynamic degrees of freedom, and is usually
phenomenological in nature. For black holes, the equation of state can be
constructed directly from (19) and (23). For $K=3$, we have that

$\displaystyle P=$ $\displaystyle\
\frac{(d-2)T}{4r_{+}}-\frac{(d-2)(d-3)b_{1}}{16\pi
r_{+}^{2}}+\frac{(d-2)\alpha_{2}b_{1}T}{2r_{+}^{3}}-\frac{(d-2)(d-5)\alpha_{2}b_{2}}{16\pi
r_{+}^{4}}$
$\displaystyle+\frac{3(d-2)\alpha_{3}b_{2}T}{4r_{+}^{5}}-\frac{(d-2)(d-7)\alpha_{3}b_{3}}{16\pi
r_{+}^{6}}+\frac{Q^{2}}{2r_{+}^{2d-4}}\ ,$ (35)

and in terms of the dimensionless variables defined above we have

$\displaystyle p=$ $\displaystyle\ \frac{t}{v}-\frac{(d-2)(d-3)b_{1}}{4\pi
v^{2}}+\frac{2\alpha b_{1}t}{v^{3}}-\frac{(d-2)(d-5)\alpha b_{2}}{4\pi
v^{4}}+\frac{3b_{2}t}{v^{5}}$ $\displaystyle-\frac{(d-2)(d-7)b_{3}}{4\pi
v^{6}}+\frac{q^{2}}{v^{2d-4}}\ .$ (36)

From the equation of state, critical points are easily identified as locations
where

$\frac{\partial p}{\partial v}=0\ ,\quad\frac{\partial^{2}p}{\partial
v^{2}}=0\ .$ (37)

The first of these conditions yields the critical temperature $t_{c}$:

$t_{c}=\frac{(d-2)\left(-4v_{c}^{10-2d}q^{2}\pi+b_{1}(d-3)v_{c}^{4}+2\alpha
b_{2}(d-5)v_{c}^{2}+3b_{3}(d-7)\right)}{2\pi\left(6\alpha
b_{1}v_{c}^{2}+v_{c}^{4}+15b_{2}\right)v_{c}}$ (38)

The second condition determines the critical volume $v_{c}$ as a solution to
the following polynomial:

$\displaystyle 0=$ $\displaystyle\ 4\pi
v_{c}^{10-2d}\left[(2d-5)v_{c}^{4}+6\alpha
b_{1}(2d-7)v_{c}^{2}+15b_{2}(2d-9)\right](d-2)q^{2}$ (39)
$\displaystyle+(d-2)\left[6\alpha((d-3)b_{1}^{2}-b_{2}(d-5))v_{c}^{6}-b_{1}(d-3)v_{c}^{8}-45b_{3}b_{2}(d-7)\right]$
$\displaystyle+(d-2)\left[6\alpha\left(5b_{2}^{2}(d-5)-9(d-7)b_{1}b_{3}\right)v_{c}^{2}-3\left(b_{2}\left(4(d-5)\alpha^{2}-15d+45\right)b_{1}\right.\right.$
$\displaystyle+\left.\left.5b_{3}(d-7)\right)v_{c}^{4}\right]$

Once the critical volume and temperature have been found, the equation of
state (4.2) determines the critical pressure $p_{c}$. As (39) is generally a
high-degree polynomial in $v_{c}$, multiple solutions may exist. We restrict
our analysis to choices of parameters for which $\\{v_{c},t_{c},p_{c}\\}$ are
all positive and real. Additional restrictions on the parameter space arise
from entropic considerations, namely the positivity of (28) which implies that

$(d-6)(d-4)v^{4}+2b_{1}\alpha(d-2)(d-6)v^{2}+3(d-2)(d-4)b_{2}>0\ .$ (40)

Since $d\geq 7$ and $\alpha>0$, this is automatically satisfied when
$b_{1}\geq 0$ and $b_{2}\geq 0$.

The Gibbs free energy $G$ and its dimensionless counterpart $g$ can likewise
be constructed. The equilibrium state of the system globally minimizes the
Gibbs free energy, and coexistence phases are easily identified as locations
where $r_{+}(G)$ is multivalued. Defining the dimensionless Gibbs free energy
as

$g=\frac{1}{\Sigma_{d-2}}\alpha_{3}^{\frac{3-d}{4}}G\ ,$ (41)

we have

$\displaystyle g=$ $\displaystyle\
\frac{q^{2}\\!\left((d-4)\left(\left(v^{4}+3b_{2}\right)d^{2}-\left(\frac{17v^{4}}{2}+\frac{39b_{2}}{2}\right)\\!d+15v^{4}+27b_{2}\right)+2b_{1}\\!\left(d-\frac{7}{2}\right)(d-2)\alpha\,v^{2}(d-6)\right)}{2(d-3)(d-2)(d-4)\left(2\alpha
v^{2}b_{1}+v^{4}+3b_{2}\right)(d-6)v^{d-3}}$
$\displaystyle-\frac{p\left(15b_{2}(d-2)(d-4)v^{d-1}+(d-6)\left(6b_{1}\alpha(d-2)v^{d+1}+v^{d+3}(d-4)\right)\right)}{4(d-6)(d-2)(d-4)\left(2\alpha
v^{2}b_{1}+v^{4}+3b_{2}\right)(d-1)}$
$\displaystyle+\frac{3b_{2}b_{3}(d-2)(d-4)v^{d-6}+6\left(b_{3}(d-6)b_{1}-\frac{b_{2}^{2}(d-4)}{2}\right)\alpha(d-2)v^{d-4}}{16\pi(d-6)v(d-4)\left(2\alpha
v^{2}b_{1}+v^{4}+3b_{2}\right)}$
$\displaystyle+\frac{\left((d-2)b_{2}\left(36-9d\right)b_{1}+5b_{3}(d-4)(d-6)+2(d-2)b_{2}\left(\alpha^{2}d-6\alpha^{2}\right)b_{1}\right)v^{d-2}}{16\pi(d-6)v(d-4)\left(2\alpha
v^{2}b_{1}+v^{4}+3b_{2}\right)}$
$\displaystyle-\frac{\alpha\left(2(d-2)b_{1}^{2}-3\,b_{2}(d-4)\right)v^{d}-b_{1}(d-4)v^{d+2}}{16\pi
v(d-4)\left(2\alpha v^{2}b_{1}+v^{4}+3b_{2}\right)}\ .$ (42)

This can be plotted parametrically against the equilibrium temperature of the
system, given by

$t=\frac{-4v^{10-2d}q^{2}\pi+4v^{6}p\pi+b_{1}(d-2)(d-3)v^{4}+\alpha
b_{2}(d-2)(d-5)v^{2}+b_{3}(d-2)(d-7)}{4\pi\left(2\alpha
v^{2}b_{1}+v^{4}+3b_{2}\right)v}$ (43)

and using the horizon radius $r_{+}$ as the parameter. The black hole size
thus serves as the order parameter characterizing distinct equilibrium states
and/or phases of the system. One can verify that these quantities agree with
known results for spherical and hyperbolic black holes with constant curvature
horizons frassino2014 . Our analysis will be predicated on the free energy and
equation of state, which provide complementary information about the location
and nature of critical points of the black hole system. We will see how moving
past the restriction of constant curvature transverse spaces markedly alters
the nature of the transitions present in these systems.

##  5 Thermodynamics of $d=7$ Exotic Black Holes

In the following sections we examine thermodynamic aspects of the general
solutions presented above, focusing on seven and eight-dimensional charged and
uncharged examples. We study the equation of state and thermodynamic free
energy in each case, and locate critical points which separate different
phases of the system. We work in the canonical ensemble where the physical
charge is fixed. Seven-dimensional black holes in third-order Lovelock gravity
do not admit horizons of non-constant curvature, so this case corresponds to
‘ordinary’ black holes that have been studied previously frassino2014 .
Nonetheless, we reexamine this case and discover new features previously
unseen for this class of black holes.

In seven dimensions, we are restricted to constant curvature horizon cross
sections. This arises from a dimensional analysis of the field equations and
the requirement that the base manifold be of constant curvature
farhangkhah2014 . This restricts our topological terms to be $b_{1}=b_{3}=\pm
1$ and $b_{2}=1$ (we may also have $b_{n}=0\ \forall n$ but we will not be
examining this case). We consider the charged and uncharged cases separately.

###  5.1 Uncharged Solutions

We begin with the uncharged solutions in seven dimensions. With $q=0$ the
equation of state becomes

$p=\frac{t}{v}-\frac{5b_{1}}{\pi v^{2}}+\frac{2\alpha
b_{1}t}{v^{3}}-\frac{5\alpha}{2\pi v^{4}}+\frac{3t}{v^{5}}\ ,$ (44)

and critical points as determined by (37) occur when

$t_{c}=\frac{10\left(b_{1}v_{c}^{2}+\alpha\right)v_{c}}{\pi\left(6\alpha
b_{1}v_{c}^{2}+v_{c}^{4}+15\right)}\ ,\quad
10b_{1}v_{c}^{6}+60\alpha^{2}b_{1}v_{c}^{2}-30\alpha
v_{c}^{4}-450b_{1}v_{c}^{2}-150\alpha=0\ .$ (45)

We first examine the case when $b_{1}=b_{2}=b_{3}=1$, where the above reduces
to

$t_{c}=\frac{10\left(v_{c}^{2}+\alpha\right)v_{c}}{\pi\left(v_{c}^{4}+6\alpha
v_{c}^{2}+15\right)}\ ,\quad 10v_{c}^{6}-30\alpha
v_{c}^{4}-\left(450-60\alpha^{2}\right)v_{c}^{2}-150\alpha=0\ .$ (46)

The critical volume in seven dimensions is therefore given by the solution to
a cubic function in $v_{c}^{2}$. Only one solution is real and positive for
$\alpha\geq\sqrt{3}$. Positivity of the entropy (28) will be automatically
satisfied for black holes of any size since all coefficients are positive. In
what follows, we fix the value of $\alpha$ to limit the size of the parameter
space, noting where variations of $\alpha$ lead to important qualitative
changes in the phase structure. Otherwise, changing $\alpha$ simply changes
the numerical values of the various critical points and bounds we find herein.
We set $\alpha=3$ so that we lie in the region of Figure 1 where all solutions
for the metric function $f$ are real and positive, and a smooth Einstein limit
exists. We are bounded only by a maximum pressure $p_{+}$, while the minimum
pressure is null:

$p_{+}=\frac{5(4\sqrt{6}-9)}{6\pi}\approx 0.211\ ,\quad p_{-}=0\ .$ (47)

In Figure 2 we show the equation of state $p(v)$, the Gibbs free energy
$g(t)$, and the phase diagram for these black holes. In Figure 2a, the
equation of state is shown for temperatures above, below, and equal to the
critical temperature $t_{c}$. Below $t_{c}$, we observe Van der Waals
oscillations characteristic of a small-large black hole phase transition.
States in the negative pressure region are not only thermodynamically unstable
(since $\partial p/\partial v>0$), but also correspond to asymptotically de
Sitter configurations (since $\Lambda>0$). The physical state of the system
instead follows the coexistence line as determined by Maxwell’s equal area
law, and is always positive in pressure due to the long tail of the $p(v)$
curve.

In Figure 2b, we plot the Gibbs free energy as a function of equilibrium
temperature at fixed pressure above, below, and at the critical pressure
$p_{c}$, using $r_{+}$ as a parameter. The radiation phase (sometimes referred
to as thermal anti-de Sitter or empty anti-de sitter) corresponds to the $g=0$
line. Along each curve, the black hole size increases as the curve is followed
from left to right. As the temperature increases, the physical state of the
system corresponds to whichever branch (including $g=0$) globally minimizes
the free energy. Above the critical pressure (as in the red curve) we observe
a Hawking-Page transition between the ‘empty’ radiation phase and a large
black hole where the red line crosses $g=0$. As the pressure decreases, a kink
eventually forms at the critical pressure where a second-order phase
transition between a small and large black hole occurs. Below $p_{c}$, a
crossing forms where the small and large black hole branches are separated by
an unstable intermediate branch, characterized by a negative specific heat. As
the temperature increases, the system follows the lower curve and a first-
order small-large transition occurs since the horizon size is discontinuous at
the crossing.

A new feature appearing in our analysis is the presence of a triple point at
sufficiently low pressure. As the pressure decreases towards zero the second-
order phase transition occurring at the crossing within the swallowtail
eventually intersects the radiation phase at $g=0$, as seen in Figure 3. At
this novel triple point, the large and small black hole phases coexist with
the radiation phase, with all three globally minimizing $g$. This is in
contrast to an ordinary triple point which consists of three phases that all
correspond to black hole states. The vertical line in Figure 3 extends upwards
and terminates at the maximum pressure $p_{+}$ (not shown). This novel triple
point was also observed in Gauss-Bonnet black holes with non-constant
curvature horizons hull2021 , and has not previously been noted in studies of
the phase structure of anti-de Sitter black holes in third-order Lovelock
gravity. Despite receiving significant attention in the hyperbolic
($\kappa=-1$) and flat ($\kappa=0$) cases, notably in farhangkhah2014 , the
spherical $\kappa=1$ case has not been analyzed as thoroughly.

Figure 2: Seven-dimensional uncharged black holes with
$b_{1}=b_{2}=b_{3}=1,\;\alpha=3$ and $q=0$. Left: $p\\!-\\!v$ isotherms with
Van der Waals oscillations occurring for temperatures below $t_{c}$. Middle:
$g\\!-\\!t$ isobars demonstrating the emergence of a swallowtail for pressures
below $p_{c}$. Right: Phase diagram, with coexistence lines (black) separating
regions where the small, large, and radiation phases occur. Figure 3: Free
energy of the seven-dimensional uncharged black hole with
$b_{1}=b_{2}=b_{3}=1,\;\alpha={3}$ and $q=0$. With $p\approx 0.027$, the
small-large transition lies on the $g=0$ line, and a triple point occurs where
the small and large black hole phases coexist with the radiation phase.

Finally, we observe that the phase structure is highly sensitive to variations
of $\alpha$. As $\alpha$ increases, the critical pressure decreases, until
$\alpha\approx 3.85$ where a turning point occurs and $p_{c}$ begins to
increase monotonically with $\alpha$. When $\alpha\approx 4.56$ the critical
pressure coincides with the maximum pressure $p_{+}$, while above this value,
no critical points occur. Despite this, one can still examine phases with
$\alpha\gtrsim 4.56$, however in the uncharged case the phenomenology is
limited to Hawking-Page phase transitions between radiation and a large black
hole. When $\alpha\gtrsim 3.87$ the novel triple point no longer occurs at any
allowed pressure. In this case the free energy at the small-large transition
always lies above $g=0$, and only a Hawking-Page transition occurs (for any
value of $p$). Finally as the value of $\alpha$ increases, the triple point
moves to the right, eventually merging with the critical point. Above this
value of $\alpha$ only a Hawking-Page transition occurs. Henceforth we fix
$\alpha=3$, as variations in $\alpha$ simply correspond to a rescaling of the
$b_{n}$ terms when $d\geq 8$.

###  5.2 Charged Solutions

We now briefly examine charged solutions with positive-curvature transverse
sections. When $q\neq 0$ and $b_{1}=b_{2}=b_{3}=1$, the critical temperature
and volume are given by

$t_{c}={\frac{10{v_{{c}}}^{8}+10\alpha{v_{{c}}}^{6}-10\pi{q}^{2}}{\pi{v_{{c}}}^{5}\left({v_{{c}}}^{4}+6\alpha{v_{{c}}}^{2}+15\right)}}\
,$ (48) $v_{c}^{12}-3\alpha
v_{c}^{10}+\left(6{\alpha}^{2}-45\right)v_{c}^{8}-15\alpha{v_{{c}}}^{6}-(9\pi{v_{{c}}}^{4}+42\pi\alpha{v_{{c}}}^{2}+75\pi){q}^{2}=0\
.$ (49)

The critical pressure is given by

$p_{c}={\frac{10{v_{{c}}}^{12}-5\alpha{v_{{c}}}^{10}+\left(10{\alpha}^{2}-90\right){v_{{c}}}^{8}-15\alpha{v_{{c}}}^{6}-18\pi{q}^{2}{v_{{c}}}^{4}-28\pi\alpha{q}^{2}{v_{{c}}}^{2}-30\pi{q}^{2}}{2\pi{v_{{c}}}^{10}\left({v_{{c}}}^{4}+6\alpha{v_{{c}}}^{2}+15\right)}}$
(50)

where $v_{c}$ will be the solution to the critical volume expression while the
temperature has been replaced by the critical temperature. The critical volume
equation can be re-expressed as the condition that $w_{1}=w_{2}$ where we
define

$w_{1}=\left(9\pi v_{c}^{4}+42\alpha\pi v_{c}^{2}+75\pi\right)q^{2},\quad
w_{2}=v_{c}^{12}-3\alpha
v_{c}^{10}+\left(6{\alpha}^{2}-45\right)v_{c}^{8}-15\alpha v_{c}^{6}.$ (51)

In this form the number of possible critical points can be easily determined
through Descartes’ rule of signs (graphically, we are concerned with the
number of intersections between $w_{1}$ and $w_{2}$). $w_{1}$ is a quadratic
in $v_{c}^{2}$ and no sign changes occur between terms, therefore no positive
roots exist ($q$ acts simply as a rescaling factor). For $w_{2}$ there are
three sign changes, so a maximum of three positive roots exist and a minimum
of one. For the case considered here where $\alpha=3$ there is only one
positive root, therefore for any value of $q$ there exists only one value of
$v_{c}$ for which $w_{1}=w_{2}$, and only one critical point exists. In what
follows the charge is fixed to $q=1$ unless charge variations are being
considered.

As with the uncharged solutions, we examine the thermodynamics behaviour
through the equation of state and free energy, as shown in Figure 4. On the
right, the same oscillatory behaviour seen in the uncharged case is present,
though the transition to the oscillatory region is now sensitive to the
charge. As the value of charge is increased, the critical temperature
decreases. In Figure 4b, the behaviour of the Gibbs free energy is
qualitatively similar to the uncharged case, but with a different
interpretation. In the canonical ensemble the physical charge is fixed, which
places a lower bound on the allowed black hole size. In the case of $d=4$
Einstein gravity for example, Reissner-Nördstrom black holes require that
$m^{2}>q^{2}$ otherwise a naked singularity forms. This constraint generalizes
to higher-dimensional Lovelock black holes, placing a lower bound on the black
hole size which depends on $\\{\alpha,q,\Lambda\\}$. Thus, Hawking-Page
transitions do not occur since the radiation phase corresponds to an $m=0$
black hole which cannot exist when $q=0$. Instead, we observe a first-order
phase transition between a small and large black hole when $p<p_{c}$, a
second-order phase transition when $p=p_{c}$, and no transitions when
$p>p_{c}$.

Figure 4: Seven-dimensional charged black holes with $b_{1}=b_{2}=b_{3}=1$,
$\alpha=3$, and $q=1$. Left: $p\\!-\\!v$ isotherms with Van der Waals
oscillations occurring for temperatures below $t_{c}$. Middle: $g\\!-\\!t$
isobars demonstrating the emergence of a swallowtail for pressures below
$p_{c}$. Right: Phase diagram, with coexistence lines (black) separating
regions where the small and large black hole phases occur.

Before moving on to eight-dimensions, we make a brief statement about the
hyperbolic case. When $b_{1}=-1,b_{2}=1,b_{3}=-1$, there are no critical
points for $\alpha=3$. Critical points are possible for values of $\alpha<3$
as seen in frassino2014 , however these values of $\alpha$ appear to
correspond to spacetimes where there is no smooth limit between the three
branches of solutions to (4.1), hence their properties and structures are not
entirely understood. Furthermore, hyperbolic geometries contain
discontinuities in both the Gibbs free energy and temperature making the
thermodynamic analysis more difficult. We therefore limit our analysis to
$b_{1}>0$ where such issues are not present.

##  6 Thermodynamics of $d=8$ Exotic Black Holes

In eight dimensions, we no longer require that $b_{1}=\kappa$,
$b_{2}=\kappa^{2}=1$ and $b_{3}=\kappa^{3}$. However some restrictions can be
placed on the $b_{n}$ by considering the character of the resulting metric
functions. In the exotic black hole solutions given by (4.1), $b_{1}$ can in
principle be chosen arbitrarily. However when $b_{1}<0$, any would-be Hawking-
Page transitions involve a change in topology in the asymptotic region, since
the spatial sections are hyperbolic in the black hole spacetime but flat in
the ‘empty’ AdS spacetime. This represents a topological transition in
addition to the thermal Hawking-Page transition. Though in a quantum setting
such transitions may be possible, we will avoid these cases so as to isolate
the purely thermal features of the phase structure. Moreover, having
$0\\!<b_{1}\\!<1$ or $b_{1}\\!>\\!1$ is also possible, but such a choice
introduces a conical deficit (or excess) in the spacetime. The metric is no
longer Ricci flat, instead resembling metrics that arise in the presence of a
global monopole charge vilenkin1994 . In the simplest case, such metrics are
sourced by $\mathcal{O}(3)$ scalar fields with Higgs potentials whose vacuum
expectation value does not vanish. Being interested in vacuum solutions with
no external fields, we wish to avoid these cases as well. As a result, we will
restrict our attention to $b_{1}=1$ throughout. $b_{2}$ and $b_{3}$ are not
constrained in this way, as they do not determine the asymptotic structure of
the spacetime.

Further restrictions can be placed on $b_{2}$ and $b_{3}$ however by
considering the behaviour of the Kretschmann invariant
$\mathcal{K}=R_{abcd}R^{abcd}$. In $d$-dimensions, its functional form is
given by

$\mathcal{K}=\left(\frac{d^{2}f(r)}{dr^{2}}\right)^{2}+\frac{2(d-2)}{r^{2}}\left(\frac{df(r)}{dr}\right)^{2}+\frac{2(d-2)(d-3)f(r)^{2}}{r^{4}}-\frac{4R[\gamma]f(r)}{r^{4}}+\frac{\mathcal{K}[\gamma]}{r^{4}}\
,$ (52)

whereas before $f(r)$ corresponds to the chosen branch of (4.1), $R[\gamma]$
is the scalar curvature of the base manifold, and $\mathcal{K}[\gamma]$ is the
Kretschmann scalar of the base manifold. $\mathcal{K}$ will diverge not only
when $f(r)$ or its derivatives diverge, but also when the following quantity
vanishes

$\displaystyle X(r)\equiv$ $\displaystyle\
81\,b_{2}-27\,b_{3}-54+\left(9\alpha-2\alpha^{3}-27\alpha_{0}\right)r^{6}$
$\displaystyle+\sqrt{4\big{(}9\,b_{2}-9-\left(\alpha^{2}-3\right)r^{4}\big{)}^{3}-\big{(}81\,b_{2}-27\,b_{3}-54+\left(9\alpha-2\alpha^{3}-27\alpha_{0}\right)r^{6}\big{)}^{2}}\
,$ (53)

which can be seen by direct substitution of (30) into (52). For all
reasonable666Reasonable here means real-valued and at most $\mathcal{O}(1)$.
choices of couplings and $b_{n}$ the metric function $f(r)$ is
$\mathcal{C}^{\infty}$, so one need only avoid choices of parameters for which
$X(r)$ vanishes for some $r$. Figure 5 shows regions in the $\\{b_{n}\\}$
parameter space where this occurs for $\alpha_{0}=0.01$ and $\alpha=3$.

Figure 5: Inadmissible choices of the parameters $\\{b_{n}\\}$ for the
8-dimensional AdS geometry, with $m\in\\{0,50,100\\}$, $b_{1}=1$,
$\alpha_{0}=0.01$ and $\alpha=3$. Left: Within the shaded region, $f(r)$ is
not defined, while on the boundary of the shaded region $\mathcal{K}$
diverges. Right: Projection onto the $b_{2}$–$b_{3}$ plane, with the exclusion
regions shaded in red.

On the left of Figure 5, we show inadmissible regions of the parameter space.
The metric function is not defined within the shaded region, while on the
boundary surface the Kretschmann invariant diverges777We have excluded the
divergence of $K$ at $r=0$ which occurs generically for any choice of base
manifold, couplings, and $b_{n}$’s.. We show two choices of $b_{2}$ and
$b_{3}$ (in black and red) demonstrating a choice where the curvature remains
finite for all $r$ (black) and one where the curvature diverges at finite $r$
(red). In the general case the singularities represented by the boundary
surface are not hidden behind an accompanying event horizon. On the right of
Figure 5, we show permissible choices of $b_{n}$ by projecting out the radial
dimension from the left figure. The shaded region then indicates that
$\mathcal{K}$ diverges at some finite $r$ for the given choice of $b_{2}$ and
$b_{3}$. It should be noted that for the $m=0$ case, the black boundary
asymptotically approaches the $b_{3}=0$ line, but does not intersect it.
Therefore, the case where $b_{2}=b_{3}=0$ is not pathological. It is also
important to note the mass-dependence of the inadmissible region. One may
appear to have, for example, a Hawking-Page transition from a black hole with
$b_{2}=0$, $b_{3}=1$ and $m=50$ to an empty spacetime, but such a transition
is not allowed since when $m=0$ (as in the radiation phase) one cannot
actually have $b_{2}=0$ and $b_{3}=1$.

In what follows, we restrict our analysis to choices that lie outside of this
region, thereby avoiding parameters for which naked singularities may form.

###  6.1 Uncharged Solutions

In $d=8$ with $q=0$ and $\alpha=3$, the maximum and minimum pressures are

$p_{+}=\frac{7(-27+12\sqrt{6})}{18\pi}\approx 0.296\ ,\quad p_{-}=0\ ,$ (54)

and the equation of state becomes

$p=\frac{t}{v}-\frac{15}{2\pi v^{2}}+\frac{6t}{v^{3}}-\frac{27b_{2}}{2\pi
v^{4}}+\frac{3b_{2}t}{v^{5}}-\frac{3b_{3}}{2\pi v^{6}}\ .$ (55)

The resulting critical volume and temperature are then given by

$\displaystyle
t_{c}=\frac{3\left(5v_{c}^{4}+18b_{2}v_{c}^{2}+3b_{3}\right)}{\pi
v_{c}\left(v_{c}^{4}+18v_{c}^{2}+15b_{2}\right)}\ ,$ (56) $\displaystyle
15v_{c}^{8}+\left(162b_{2}-270\right)v_{c}^{6}+\left(297b_{2}+45b_{3}\right)v_{c}^{4}-\left(810b_{2}^{2}-486b_{3}\right)v_{c}^{2}+135b_{2}b_{3}=0\
.$ (57)

While the topological parameters are independent of each other, thermodynamic
considerations lead to further constraints on their values. Re-arranging the
equation of state (4.2) as

$t\rightarrow t(p,v,b_{2},b_{3})=\frac{2\pi
pv^{6}+15v^{4}+27v^{2}b_{2}+3b_{3}}{2\pi v\left(v^{4}+6v^{2}+3b_{2}\right)}\ $
(58)

reveals that negative values of $b_{2}$ lead to a divergent temperature due to
the $(v^{4}+6v^{2}+3b_{2})$ term in the denominator. We therefore restrict our
analysis to the case where $b_{2}\geq 0$. This $(v^{4}+6v^{2}+3b_{2})$ term is
also present in the free energy $g$, which then diverges along with the
temperature. From equation (57) we can determine the possible number of
positive real $v_{c}$ solutions through Descartes’ rule of signs, which are
summarized in Table 1 below.

| $b_{2}=0$ | $b_{2}>0$
---|---|---
$b_{3}=0$ | 1 | 3,1
$b_{3}<0$ | 1 | 3,1
$b_{3}>0$ | 2,0 | 4,2,0
Table 1: Possible number of positive real roots to 57 from Descartes’ rule of
signs applied to the topological parameters $b_{2}$ and $b_{3}$.

We begin with the case where $b_{2}=0$ and $b_{3}=-1$. This choice results in
thermodynamic behaviour similar to the seven-dimensional case, with small-
large transitions appearing generically below the critical temperature and
pressure, as shown in Figure 6. Note that there is no Hawking-Page transition
at low temperatures, and that $G\rightarrow 0$ as $T\rightarrow 0$.

Criticality is also present when $b_{2}=0$ and $b_{3}=0$. The free energy in
this case is qualitatively similar to that of Figure 6b, and the $p-v$ curves
likewise follow Figure 6a. The critical pressure and temperature both increase
as $b_{3}$ increases.

Figure 6: Eight-dimensional uncharged black holes with $b_{1}=1$, $b_{2}=0$,
$b_{3}=-1$, and $\alpha=3$. Left: $p\\!-\\!v$ isotherms with Van der Waals
oscillations occurring for temperatures below $t_{c}$. Center: $g\\!-\\!t$
isobars demonstrating the emergence of a swallowtail for pressures below
$p_{c}$. Right: Phase diagram, with coexistence lines (black) separating
regions where the small and large black hole phases occur.

When $b_{2}=0$ and $b_{3}>0$, two real and positive solutions for $v_{c}$ will
exist for $b_{3}<\frac{324}{25}$. When $b_{3}\geq 1$, the novel triple point
emerges (seen also in the uncharged seven-dimensional case above) with notable
qualitative differences. Increasing $b_{3}$ likewise increases the temperature
and pressure at which this triple point occurs. Eventually, when $b_{3}\approx
6$, the triple point is no longer present as the novel triple point merges
with the smaller of the two critical points and only the Hawking-Page
transition is seen.

Next we consider the case where $b_{2}\neq 0$. When $b_{2}=1$ and $b_{3}=0$,
Van der Waals behaviour is observed corresponding to a small-large black hole
transition, though no triple point occurs at any temperature/pressure. When
$b_{3}>0$ instead, we observe the same qualitative behaviour noted above for
$b_{2}=0$ and $b_{3}>0$: the existence of a triple point below a certain value
of the parameter (in this case $0<b_{3}<0.8$) , and Hawking-Page transitions
when $b_{3}>0.8$.

###  6.2 Charged Solutions

When $d=8$ and $q\neq 0$ the equation of state (4.2) becomes

$p=\frac{t}{v}-\frac{15}{2\pi v^{2}}+\frac{6t}{v^{3}}-\frac{27b_{2}}{2\pi
v^{4}}+\frac{3b_{2}t}{v^{5}}-\frac{3b_{3}}{2\pi v^{6}}+\frac{q^{2}}{v^{12}}\
,$ (59)

with the critical temperature and volume given by

$t_{c}=-\frac{3\left(-5v_{c}^{10}-18b_{2}v_{c}^{8}-3b_{3}v_{c}^{6}+4\pi
q^{2}\right)}{\pi v_{c}^{7}\left(v_{c}^{4}+18v_{c}^{2}+15b_{2}\right)}\ ,$
(60) $\displaystyle
15v_{c}^{14}+\left(162b_{2}-270\right)v_{c}^{12}+\left(297b_{2}+45b_{3}\right)v_{c}^{10}-\left(810b_{2}^{2}-486b_{3}\right)v_{c}^{8}+135b_{2}b_{3}v_{c}^{6}$
$\displaystyle-\left(132\pi v_{c}^{4}+1944\pi v_{c}^{2}+1260\pi
b_{2}\right)q^{2}=0$ (61)

In principle, a systematic determination of the root structure of the
polynomial above can be done using Descartes’ rule of signs; however, the
presence of $b_{3}$ and $q$-dependent terms means such an analysis quickly
becomes cumbersome. Instead we proceed on a case-by-case basis, as only a few
examples are required to demonstrate the salient features of charged $d=8$
black holes.

As with the uncharged case, we begin with the spherical base manifold for
which $b_{n}=1$. In this case, we observe only a small-large transition
terminating at a critical point $\\{t_{c},p_{c}\\}$ for all values of the
charge $q$. This behaviour is identical to what is shown in Figure 4 for the
$d=7$ case. As $q$ increases, $t_{c}$ and $p_{c}$ decrease, until they
eventually achieve the $q=0$ values shown in Figure 6. That there can only be
a small-large transition for 8-dimensional charged black holes is easily seen
by separating the charge-dependent and charge-independent portions of (6.2),
and studying the number of intersections of the resulting functions. Let

$F_{1}=F_{2}$ (62)

with

$\displaystyle F_{1}$ $\displaystyle\equiv\left(132\pi v_{c}^{4}+1944\pi
v_{c}^{2}+1260\pi\right)q^{2}$ (63) $\displaystyle F_{2}$ $\displaystyle\equiv
15v_{c}^{14}-108v_{c}^{12}+342v_{c}^{10}-324v_{c}^{8}+135v_{c}^{6}\ .$

Plotting $F_{1}$ and $F_{2}$ for varying values of $q$ reveals that for any
finite value of the charge there is a unique value of $v_{c}$ which satisfies
(62), as displayed in Figure 7.

Figure 7: Plots for $F_{1}$ and $F_{2}$ as a function of $v_{c}$ for varying
values of $q$. Left: $q=0.1$, Center: $q=0.5$, Right: $q=1$.

As in the uncharged case, we plot the equation of state $p(v)$ and free energy
$g(t)$, searching for Van der Waals oscillations in the $p-v$ plane and the
formation of a swallowtail in the $g-t$ plane. Note that even though critical
points may occur above $g=0$, the radiation phase is inaccessible to the
system since the charge is fixed in the canonical ensemble, so no Hawking-Page
transitions will occur in any of the cases we examine below.

Beginning with $b_{2}=0$, when $b_{3}\leq 0$ we observe a single critical
point and the corresponding Van der Waals-like transition between a large and
small black hole, for all values of $q$. Only quantitative changes in the
critical temperature and volume occur when $q$ is varied. When $b_{3}>0$, a
triple point may occur for very specific values of $q$ and $b_{3}$. For
example, when $b_{3}=0.01$, the standard small-intermediate-large black hole
triple point occurs when $0<q\lesssim 0.0069$. When $q\gtrsim 0.0069$ the
triple point vanishes and only a small-large transition occurs. On the other
hand, when the charge is fixed the existence of the triple point is lower
bounded by the parameter $b_{3}$. For example, when $q=0.05$ we must have
$b_{3}>0.4$ for a triple point to occur. This is shown in Figure 9.

Next we consider the case where $b_{2}>0$. For any choice of $b_{2}>0$ there
is no choice of $q$ and $b_{3}<0$ that yields a triple point. The only
phenomena observed are small-large phase transitions. This is also the case
when $b_{3}=0$. When $b_{3}>0$, we have the possibility of a small-
intermediate-large triple point. As above, the presence of this triple point
is highly sensitive to the values of $q$ and $b_{2}$. If $b_{2}=0.1$ and
$b_{3}=0.1$, the triple point only occurs for $0<q<0.006$. If $b_{3}$ is
increased while $b_{2}$ is kept fixed, the triple point will occur at a larger
value of $q$. Correspondingly, the location of the triple point occurs at a
larger temperature $t_{c}$ and pressure $p_{c}$. Eventually, if both $b_{3}$
and $q$ are sufficiently large, the triple point moves above the maximum
allowed pressure $p_{max}$, and only a small-large transition is physically
allowed. Therefore as long as the maximum pressure bound is not violated, a
triple point can occur for any choice of $b_{2}>0$ and $b_{3}>0$ provided the
black hole has the correct charge.

The behaviour for fixed $b_{3}$ and varying $b_{2}$ and $q$ is qualitatively
different. As a concrete example, take $b_{3}=0.1$ and increase $b_{2}$ and
$q$ from 0.1 and 0 respectively. A triple point occurs as long as
$0.1<b_{2}<0.2$ and $0<q\lesssim 0.003$. If both $b_{2}\geq 0.2$ and $q\gtrsim
0.003$ only a small-large transition occurs. Generically, when the charge
decreases the temperature and pressure at which the triple point occurs will
increase. In all cases, analytic solutions to the locations of the
triple/critical points are not available, so the accuracy in the bounds
presented are constrained by the choice of numerical solver and precision
used.

The equation of state, free energy, and phase diagram in the presence of a
triple point are shown in Figure 9. The triple point occurs at the
intersection of the small, intermediate, and large black hole branches of the
$g-t$ diagram (indicated also by a red dot in the phase diagram). At this
point, three distinct black hole states with different sizes all globally
minimize the free energy at the same temperature. Figure 9c contains the phase
diagram, which displays new features compared to the uncharged case. The
rightmost branch distinguishes the large and intermediate black hole phases
and terminates at a critical point, beyond which the system is in a
‘superfluid’ phase where the two states are no longer distinguishable. The
nearly vertical branch separates the intermediate and small black hole phases,
and extends upward until it terminates at the maximum allowed pressure
$p_{max}$. This is conjectured to always occur before any critical point forms
regardless of parameter choice, though a lack of analytic solutions for the
location of the critical points prevents us from excluding the possibility of
a second critical point occurring below $p_{max}$. Above the critical
temperature/pressure which terminates the intermediate-large transition, the
two phases are no longer distinguishable (there is no discontinuity in the
horizon size). However, we still observe a phase transition between this
‘merged’ intermediate-large black hole branch and the small black hole branch.

Figure 8: Eight-dimensional charged black holes with $b_{1}=b_{2}=b_{3}=q=1.$
Left: $p-v$ diagram showing oscillation. Center: $g-t$ diagram with standard
swallow tail structure. Right: Phase diagram with coexistence line showing the
separation of large and small black holes which terminates at the critical
point.

Figure 9: Eight-dimensional charged black hole with
$b_{1}=1,\;b_{2}=0.1,\;b_{3}=0.5,\;q=0.05$. Left: $p-v$ diagram for $t\approx
0.415$ showing two oscillations. Center: $g-t$ diagram for $p\approx 0.19$
showing the intersection of the two swallow tails. Right: Phase diagram
showing coexistence lines that all join together at the red triple point. The
intermediate-large line will terminate at the critical point. The
intermediate/small coexistence line will terminate at the maximum pressure of
the spacetime.

##  7 Discussion

Higher curvature (and thus, necessarily higher-dimensional) generalizations of
Einstein gravity admit the existence of black hole solutions with ‘exotic’
horizon geometries. In particular, Lovelock gravity allows for black holes
which are distinguished by non-constant curvature horizon cross-sections. This
is in stark contrast to vacuum solutions in $d=4$ Einstein gravity, which are
constrained by Birkhoff’s theorem to have transverse metrics isometric to
$\mathcal{S}^{2}$ in the case of spherical symmetry. In this work, we have
examined the phase structure of such black hole solutions in third-order
Lovelock gravity, focusing on both $U(1)$ charged and uncharged asymptotically
AdS solutions in 7 and 8 dimensions and in an extended phase space where the
cosmological constant acts as a thermodynamic pressure. Our work can be seen
as a natural extension of some preliminary efforts at understanding the
thermodynamic properties of these exotic black hole solutions as seen in
farhangkhah2014 ; hull2021 .

The generalization to third-order Lovelock gravity (and by necessity $d\geq
7$) retains much of the qualitative features of exotic black holes in second-
order Gauss-bonnet gravity in $d=5$. Nevertheless we observed and accounted
for a wide variety of thermodynamic phenomena which are generic to
asymptotically AdS black holes, and found that novel triple point phenomena as
seen in hull2021  persist in higher-order Lovelock gravity. We also clarified
the interpretation of earlier results in the case of $d=5$ exotic black holes
and $d=7$ constant curvature black holes, and analyzed the space of physically
reasonable exotic black hole solutions.

General ‘exotic’ black hole solutions in third-order Lovelock gravity depend
on a set of arbitrary constants $\\{b_{1},b_{2},b_{3}\\}$ characterizing the
base manifold, as well as the Lovelock coupling constants
$\\{\alpha_{0},\alpha_{2},\alpha_{3}\\}$. We obtained a discriminant condition
for the metric function $f(r)$ that establishes a maximum and minimum allowed
pressure, which all branches of solution are constrained by. The analysis
focused on the Einstein branch of solutions, which has a smooth limit to
Einstein gravity when $\alpha_{2},\alpha_{3}\rightarrow 0$. The topological
parameters $\\{b_{n}\\}$ were treated as constants which can in principle take
any values in $\mathbb{R}$. To our knowledge there is currently no known
prescription for systematically determining all allowed combinations of
topological parameters in third-order Lovelock gravity. Nevertheless, we were
able to rule out a large portion of the parameter space by studying how the
formation of naked singularities is sensitive to the choice of transverse
geometry. We found that most choices of transverse space are pathological,
containing a divergence in the Kretschmann scalar at a finite radius outside
of the event horizon. This is not surprising as the allowed exotic solutions
are already severely constrained in Gauss-Bonnet gravity Dotti_2005 .

In seven dimensions, we found that uncharged black holes with a spherical base
manifold contain a novel triple point akin to the one found in hull2021 ,
which was not previously seen in the analyses of frassino2014  and
farhangkhah2014 . This triple point is found to be sensitive to the rescaled
Gauss-Bonnet coupling $\alpha$, terminating at the critical point once
$\alpha$ reaches a certain value (after which only a Hawking-Page transition
occurs). Furthermore, small-large black hole phase transitions are forbidden
as they require $\alpha<3$ which precludes the existence of a smooth Einstein
limit (see Figure 1). When charge is present, the triple point vanishes and
only small-large transitions occur with a single critical point. We clarify
that although Hawking-Page transitions were reported previously for this case
farhangkhah2014 , the canonical (fixed-charge) ensemble does not allow for
such transitions, even if they appear energetically favourable.

In eight dimensions, black holes without charge exhibit three distinct phase
transition behaviours: small-large transitions, Hawking-Page transitions, or
triple point phenomena, depending on the choice of topological parameters. For
charged black holes, the Hawking-Page transition no longer occurs and we only
observe a small-large transition or a triple point. Surprisingly, new types of
phase transitions are not seen even with the inclusion of a new topological
parameter $b_{3}$. One might expect a new branch of phase transitions (or even
a quadruple point with two intermediate branches) to emerge in $d=8$, because
the equation of state contains new terms in higher inverse powers of $r$. It
turns out, however, that the constraints present do not allow for choices of
parameters which would produce the necessary oscillations/branch structure for
such phenomena to occur.

Along the way, we have found solutions for black holes with vanishing mass
parameter which nevertheless possess horizons and non-trivial spacetime
singularities. These can be found directly from (30) by appropriate choice of
parameters. Investigating the thermodynamic properties and geometric structure
of these solutions would be an interesting future endeavour. Another ambitious
task would be to examine exotic black holes in asymptotically de-Sitter
spaces, where the notion of thermodynamic equilibrium is more subtle. This is
an area which has been gaining attention in recent years sekiwa2006 ;
zhang2014 ; haroon2020 ; mbarek2019 ; simovic2021  with many new thermodynamic
features being uncovered for a wide class of black holes. Finally,
understanding fully how to constrain the topological parameters would be a
significant step towards determining exactly which types of exotic black holes
deserve further investigation.

## Acknowledgments

B.H. is supported in part by the Natural Sciences and Engineering Research
Council of Canada. F.S is supported by the Australian Research Council. We are
grateful to Robert B. Mann for a number of helpful discussions.

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# How BERT Speaks Shakespearean English?
Evaluating Historical Bias in Contextual Language Models

Miriam Cuscito
Università degli Studi di Cassino e del Lazio Meridionale
<EMAIL_ADDRESS>Alfio Ferrara  Martin Ruskov
Università degli Studi di Milano
<EMAIL_ADDRESS>

###### Abstract

In this paper, we explore the idea of analysing the historical bias of
contextual language models based on BERT by measuring their adequacy with
respect to Early Modern (EME) and Modern (ME) English. In our preliminary
experiments, we perform fill-in-the-blank tests with 60 masked sentences (20
EME-specific, 20 ME-specific and 20 generic) and three different models (i.e.,
BERT Base, MacBERTh, English HLM) 111https://huggingface.co/bert-base-uncased,
https://huggingface.co/emanjavacas/MacBERTh,
https://huggingface.co/dbmdz/bert-base-historic-english-cased. We then rate
the model predictions according to a 5-point bipolar scale between the two
language varieties and derive a weighted score to measure the adequacy of each
model to EME and ME varieties of English.

##  1 Introduction

Contextual language models (CLMs) are deep neural language models which create
contextualised word representations, in the sense that the representation for
each word depends on the entire context in which it is used. That is to say,
word representations are a function of the entire input sentence. Such models
are usually pre-trained on large textual corpora and designed to have high
predictive capabilities. This makes them closely tied to the domains on which
they were trained and dependent on the infrastructure upon which they are
based.

The presence of various biases in CLMs has been extensively studied, typically
with the aim of proposing effective mitigation strategies de Vassimon Manela
et al. (2021); Ahn and Oh (2021); Mozafari et al. (2020). However, there are
instances where the bias in certain CLMs is not necessarily negative. This is
particularly true when the bias manifested in the language reflects its socio-
temporal context. This bias could be advantageous for tasks that demand this
socio-temporal staging Corfield (2022); Diaz-Faes et al. (2023).

In this paper, we explore the idea of analysing the bias by focusing on the
major syntactic, semantic, and grammatical differences between two varieties
of the English language: Early Modern (EME) and Modern (ME). More precisely,
we propose a method and a measure of adequacy to test the adherence of CLMs to
the natural language variety of interest. In particular, we assess the level
of diachronic bias of three CLMs: Bert-Base-Uncased (BERT Base) Devlin et al.
(2019); MacBERTh Manjavacas and Fonteyn (2022); and Bert Base Historic English
(English HLM). In our preliminary experiments, we perform tests with 60 masked
questions in which the models have the task to predict the masked word in the
sentence. We then rate the proposed responses according to a 5-point bipolar
scale between the two language variants and derive a weighted score from the
response probabilities and their respective scores on the scale.

These results, although preliminary, might suggest a method applicable in the
digital humanities when CLMs are employed for the analysis of historical
corpora.

##  2 Related Work

If it is true that language shapes culture while it is shaped by it Boroditsky
(2011), language models in general – and CLMs in particular – constitute a
still partially covered mirror of this dual relationship. Not only can a CLM
be tested based on its level of representativeness of the language to
determine its reliability, but also it can tell us about linguistic, social,
and historical phenomena that concern the culture tied to that specific
language. In other words, a CLM could be a valuable tool towards the expansion
of the broader social knowledge of a given culture, rightfully becoming part
of the basic tools of Cultural Analytics discussed by Manovich Manovich
(2020). According to Bruner’s Bruner (1984) pragmatic-cultural perspective,
learning a language also means learning the cultural patterns associated with
it. Similarly, analysing the language in its various realisations would mean
having the opportunity to visualise the underlying cultural patterns.

Moreover, CLMs can be highly beneficial also for philological van Lit (2019),
pragmatic Ruskov (2023), critical Diaz-Faes et al. (2023), and literary work
Piper et al. (2023). However, the effectiveness of these models depends on
their ability to adapt to language specificity in its historical dimension.
This is typically achieved by training models on historical text corpora.
However, the difficulty of accessing large historical documentary collections
means that the models available are still few and requires verifying whether
they adapt effectively to the historical linguistic context.

BERT is a foundational contextual language model (CLM) which to date is the
most widely adopted Montanelli and Periti (2023). A number of studies have
explored different forms of bias in BERT de Vassimon Manela et al. (2021); Ahn
and Oh (2021); Mozafari et al. (2020). Three BERT-based CLMs are of particular
interest for our study:  (i) Bert-Base-Uncased Devlin et al. (2019), created
from a corpus of texts from Wikipedia and BookCorpus and a model of
contemporary language, which we use as a control condition in our experiment;
(ii) MacBERTh Manjavacas and Fonteyn (2022), pre-trained on texts from 1500 to
1950; and  (iii) Bert-Base-Historic-English, pre-trained on contemporary texts
and fine-tuned on historical texts from the 19th century to the present  .

##  3 Method

To evaluate the adequacy of CLMs on a test set, we define a temporal valence
task consisting of a collection of test sentences, each with a masked token
(i.e., word). This is a typical fill-in-the-blank task, where the models are
required to predict the masked token. Formally, we consider the following
three sets:  (i) we denote with $\mathcal{S}$ the set of all test _sentences_
,  (ii) with $\mathcal{V}$ we denote a set of _vocabulary_ words, and  (iii)
with $\mathcal{T}=\\{-1,-0.5,0,0.5,1\\}\subset\mathbb{R}$, we denote a 5-point
bipolar _temporal valence scale_ , where $-1$ represents the farthest
historical period and $1$ the closest to today.

With the above notation, for each of the masked sentences (denoted as
$s\in\mathcal{S}$), we define a function
$\rho:\mathcal{S}\rightarrow\mathcal{T}$ representing the _sentence temporal
valence score_. This function indicates the period from which the masked
sentence is typical.

Then, we calculate a _token-in-sentence temporal valence score_
$\sigma:\mathcal{V}\times\mathcal{S}\rightarrow\mathcal{T}$, indicating the
score of a token substituting the sentence mask.

The mentioned _temporal valence scores_ are assigned arbitrarily according to
the research hypotheses. Taking this study as an example, the criterion used
to determine each score was the degree of alignment of certain sentences or
tokens with a specific historical period on a philological-linguistic basis.
Scholars wishing to delve into language study using this methodological
approach can selectively choose the score to assign to their test set based on
their specific research needs. The versatility of the proposed methodology is
evident in its adaptability to a diverse array of fields of interest. This
flexibility enables researchers to seamlessly integrate personalized metrics,
ensuring a tailored approach to analysis without undermining the inherent
consistency of the results.

As an example of _temporal valence score_ , given EME (Early Modern English)
as the farthest period (i.e., $-1\in\mathcal{T}$) and ME (Modern English) as
closest (i.e., $1\in\mathcal{T}$), if we consider the sentence $s_{1}=$ _“Why
wilt [MASK] be offended by that?”_ we have $\rho(s_{1})=-1$ as $s_{1}$ is a
representative sentence for EME, and $\sigma(``thou",s_{1})=-1$, because in
this context “thou” is indicative for EME. On the other hand,
$\sigma(``not",s_{1})=0$, because “not” is neutral regarding the two language
varieties.

Given a model $m$, for the masked token in each sentence ($s\in S$), we have
the set of $\\{w_{1},w_{2},\dots,w_{n}\\}\subset\mathcal{V}$ of $n$ words
predicted by $m$ for $s$, that are associated with the vector of corresponding
probabilities $\mathbf{p}_{m}=(p(w_{1}),p(w_{2}),\dots,p(w_{n}))^{T}$.

For this set, we exploit the score function $\sigma$ in order to define a
token-in-sentence temporal valence score vector $\mathbf{x}_{m}$ for $m$ given
the sentence $s$, that is
$\mathbf{x}_{m}=(\sigma(w_{1},s),\sigma(w_{2},s),\dots,\sigma(w_{n},s))^{T}$.

This allows us to define the _bias_ of a model regarding the sentence as a
weighted score:

$\beta(m,s)=\mathbf{x}_{m}^{T}\mathbf{p}_{m}$

Then, we proceed to define the _domain adequacy_ of a model with respect to a
sentence $s$ as

$\delta(m,s)=1-\frac{1}{2}\mid\rho(s)-\beta(m,s)\mid$

based on the difference between the sentence temporal valence score $\rho(s)$
and the model bias $\beta(m,s)$.

An example of three sentences from different periods is provided in Tables 1,
2 and 3, which show the corresponding values for $\rho$, $p$, $\sigma$,
$\beta(m,s)$ and $\delta(m,s)$.

BERT Base token $p$ $\sigma$ `thou` 0.712 -1.0 `you` 0.101 0.0 `i` 0.085 0.0
`she` 0.055 0.0 `he` 0.048 0.0 $\beta$ -0.712 $\delta$ 0.856 MacBERTh token
$p$ $\sigma$ `thou` 0.987 -1.0 `not` 0.008 0.0 `you` 0.004 0.0 `ye` 0.001 -1.0
`he` 0.000 0.0 $\beta$ -0.988 $\delta$ 0.994 English HLM token $p$ $\sigma$
`not` 0.639 0.0 `thou` 0.303 -1.0 `never` 0.031 0.0 `[UNK]` 0.022 0.0 `ever`
0.005 0.0 $\beta$ -0.303 $\delta$ 0.652

Table 1: Scores of the models for “Why willt [MASK] be offended by that?”
(temporal valence $\rho=-1$)

BERT Base token $p$ $\sigma$ `here` 0.924 0.0 `back` 0.066 0.5 `there` 0.004
-0.5 `forth` 0.003 -0.5 `out` 0.003 1 $\beta$ 0.032 $\delta$ 0.984 MacBERTh
token $p$ $\sigma$ `hither` 0.740 -1.0 `down` 0.170 0.0 `thus` 0.045 -0.5 `in`
0.025 0.0 `again` 0.020 0.0 $\beta$ -0.762 $\delta$ 0.619 English HLM token
$p$ $\sigma$ `here` 0.691 0.0 `back` 0.194 0.5 `again` 0.051 0.0 `in` 0.034
0.0 `hither` 0.031 -1.0 $\beta$ 0.066 $\delta$ 0.967

Table 2: Scores for “Have you come [MASK] to torment us before the time?”
($\rho=0$)

BERT Base token $p$ $\sigma$ `orientation` 0.720 1.0 `misconduct` 0.112 1.0
`minorities` 0.067 1.0 `partners` 0.052 1.0 `harassment` 0.048 1.0 $\beta$
1.000 $\delta$ 1.000 MacBERTh token $p$ $\sigma$ `##ists` 0.493 -0.5
`offenders` 0.165 1.0 `characters` 0.130 0.5 `drunkards` 0.117 0.0
`delinquents` 0.095 0.5 $\beta$ 0.031 $\delta$ 0.516 English HLM token $p$
$\sigma$ `to` 0.319 0.0 `must` 0.294 0.0 `may` 0.187 0.0 `would` 0.104 0.0
`should` 0.096 0.0 $\beta$ 0.000 $\delta$ 0.500

Table 3: Scores for “Should men who are known sexual [MASK] be given a
platform?” ($\rho=1$)

##  4 Evaluation

We test our metrics with three BERT-based linguistic models we consider
relevant for the varieties of the English language of interest: _(i)_ Bert-
Base-Uncased, _(ii)_ MacBERTh, and _(iii)_ Bert-Base-Historic-English. In
accordance with the objectives of this study, the choice of models reflects a
specific interest in language; therefore, they can be replaced to best fit any
other specific interest in diachronic language analysis. For the test we used
60 word-masked sentences, specifically created for this study. To create the
test set, we relied on different types of written language: contemporary
standard, journalistic language, social media non-standard, and Early Modern
language.

The elements to be masked were selected based on their belonging to specific
word classes known to have suffered more exposure to the diachronic variation
of the English language: pronouns, verbs, adverbs, adjectives, and nouns. Of
the 60 sentences provided in Appendix, 20 are selected to be suggestive for
the EME variety of English, further 20 – as suggestive for ME, and final 20
are generic. Once the test set was complete, a temporal valence score was
assigned to each sentence (see $\rho$ in Section 3) based on their level of
chronological markedness.

The test set was administered to the three CLMs, and the suggested words with
their probability were collected. The resultant vocabulary was marked
independently from the models that provided it by setting the token-in-
sentence temporal valence score (i.e. $\sigma$) to each word, based on an
estimation of proximity of the token’s meaning to a certain linguistic variety
in the context in which it appeared. Notably, during this phase, our decision
was to work on a sentence level (contextually) rather than on a set level
(globally). The method proved highly effective in avoiding the risk of
semantic flattening, given that almost every word has shown some level of
contextual semantic specificity if taken contextually rather than globally. An
example is the pronoun you in “fare you well, sir”, which is globally neutral
and yet acquires a strong diachronic value if evaluated in its context, in
which it appears to be utmost archaic.

Once $\beta$ and $\delta$ were calculated, we proceeded with the analysis of
the data and the collection of results. The distribution of the bias score
$\beta$ and the domain adequacy score $\delta$ for the sentences in the three
groups (i.e., EME, Neutral, and ME) is shown in Figures 1 and 2, respectively.

Figure 1: Distribution of bias $\beta(m,s)$ of the three models with respect
to the three test sets. Figure 2: Distribution of the domain adequacy
$\delta(m,s)$ of the three models with respect to the three test sets.

Figure 1 shows that for all three test sets, MacBERTh is most aligned with
EME, whereas BERT Base is always most aligned as ME. Historical BERT shows a
tendency towards a more neutral language than the other two models in marked
sentences, whilst surprisingly it aligns to ME in neutral sentences.

Figure 2 shows that MacBERTh has best domain adequacy for EME, and BERT Base
has best domain adequacy for ME. In the case of the neutral test set, domain
adequacy is no less informative. Although the sentences do not inherently
carry their expectations regarding language, models appear consistently well-
suited to a neutral context, and none of them pushes for strong specificity of
their corresponding trained domain. In effect this leaves the sentences close
to their original neutrality

This preliminary study provides an illustration of the nature and functioning
of the CLMs predictive behaviour. The presence or absence of markedness in the
sentences enables all three CLMs to select the type of element which best fits
the co-text. So, while for diachronically marked sentences, models without
training in that domain attempted to suggest probable solutions, sometimes
resulting in a form of linguistically inconsistent mimicry, in unmarked
sentences, the models perform exceptionally well, and linguistic inaccuracies
are rare.

##  5 Conclusion

Both notions of _bias_ ($\beta$) and _domain adequacy_ ($\delta$) provide
important insights of the nature of the models. The first, $\beta$, indicates
a tendency in terms of temporal valency. In other words, the interpretation of
its value should be considered within the context of the specific dichotomy of
language varieties. On the other hand, $\delta$ reflects the adequacy for an
individual language variety. It successfully captures model tendencies when
completing historically predetermined sentences. However, it falls short in
capturing the preferred language variety when completing temporality-neutral
sentences.

The notion that LMs can serve as a window into the history of a population is
not new, but there is a growing interest in exploring the relationships
between these models and the sociolinguistic and sociocultural contexts. It is
equally imperative to establish a procedural framework to address the lack of
evaluative methods for these models, as previously hinted at in this text.

Within this evaluation, we created a dedicated test set for each model under
scrutiny, drawing upon approaches used for evaluation of bias in CLMs. In
creating our test sets, we built our sentences both on logical-semantic and
logical-syntactic tasks. Future work could try to create a test set for model
interrogation that is culture-oriented, delving into socio-culturally
significant elements such as customs, historical events, and attitudes towards
social groups – elements recognised as belonging to social knowledge.

This study aims not only to propose a methodology for assessing language
models but also to put forth hypotheses for expanding the available tools to
humanities scholars interested in studying complex socio-cultural phenomena
with an approach which begins by interpreting textual clues and inferring
their connections to reality.

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##  Appendix A Test Set

For transparency and reproducibility purposes, the following anonymous link
contains the complete test set with the corresponding values produced during
evaluation:
https://tinyurl.com/bert-shakespearean