aainstitutetext: Department of Software Engineering, Faculty of Engineering
and Architecture,
Ankara Science University, 06200 Ankara, Turkeybbinstitutetext: Department of
Physics, Faculty of Science and Letters,
Istanbul Technical University, Maslak 34469 Istanbul, Turkeyccinstitutetext:
Department of Physics, Faculty of Arts and Sciences,
Sinop University, 57000, Sinop, Turkeyddinstitutetext: Department of Physics,
Faculty of Arts and Sciences,
Middle East Technical University, 06800, Ankara, Turkeyeeinstitutetext:
Department of Physics Engineering, Faculty of Engineering and Natural
Sciences,
Istanbul Medeniyet University, 34000, Istanbul, Turkey

# Scaling symmetry, Smarr relation, and the extended first law in lower-
dimensional Lovelock gravity

Gökhan Alkaç b , Gökçen Deniz Özen c , Hikmet Özşahin d , Gün Süer e ,
Mustafa Tek<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Recently, it was discovered that lower-dimensional versions of Lovelock
gravity exist as scalar-tensor theories that are examples of Horndeski
gravity. We study the thermodynamics of the static black hole solutions
satisfying $g_{tt}g_{rr}=-1$ in these theories up to cubic order through
Euclidean methods. Considering solutions with spherical, planar and hyperbolic
event horizons ($k=+1,0,-1$), we show that the universality of the
thermodynamics for planar black holes ($k=0$) and the extended 1st law that
include the variation of the couplings together with their associated
potentials hold also in lower dimensions. We find that in $D=4,6$ where the
2nd- and the 3rd-order Lovelock Lagrangians are boundary terms respectively,
the Smarr relation is modified since the entropy is not a homogenous function
in these dimensions. We also present a derivation of the Smarr relation and
its modified version based on the global scaling properties of the reduced
action that is used to obtain the solutions consistently. Unlike the other
hairy black hole solutions that have been analyzed before, despite the terms
in the reduced action that break the scaling symmetry, the derivation still
follows from a conserved Noether charge.

###### Keywords:

Black Holes, Classical Theories of Gravity

††arxiv: 2308.00620

.

##  1 Introduction

For a better understanding of Einstein’s general relativity (GR), it is of
crucial importance to find out which features are special to it and which ones
can be realized in some alternatives. Among the theories that can be derived
from a Lagrangian of the form $\mathcal{L}[g_{\mu\nu},R^{\mu}_{\
\nu\rho\sigma}]$, which naturally provides a generalization to Einstein tensor
(symmetric and covariantly conserved) that can be consistently coupled to a
covariantly conserved energy-momentum tensor, the Lovelock gravity
Lanczos:1938sf ; Lovelock:1971yv ; Lovelock:1972vz  (see Padmanabhan:2013xyr
for a review of important properties) shares the following properties:

  1. i)

The field equations contain at most the second derivative of the metric.

  2. ii)

It possesses a unitary massless spin-2 graviton around _any_ of its constant
curvature vacua Sisman:2012rc .

  3. iii)

Its metric and Palatini formulations are equivalent when the torsion is zero
Exirifard:2007da .

The Lagrangian of Lovelock gravity is given by

$\mathcal{L}=\sum_{m}c_{m}\mathcal{L}_{m},$ (1)

where the $m$-th order Lovelock Lagrangian is defined as

$\mathcal{L}_{m}=\frac{1}{2^{m}}\delta^{\mu_{1}\nu_{1}\cdots\mu_{m}\nu_{m}}_{\rho_{1}\sigma_{1}\cdots\rho_{m}\sigma_{m}}R^{\rho_{1}\sigma_{1}}_{\
\ \ \ \mu_{1}\nu_{1}}\cdots R^{\rho_{m}\sigma_{m}}_{\ \ \ \ \mu_{m}\nu_{m}}.$
(2)

It is a boundary term when111For this reason, $D=2m$ is called the critical
dimension for the $m$-th order Lagrangian $\mathcal{L}_{m}$. $D=2m$ and
vanishes identically in $D<2m$. Therefore, one can have a more general theory
with the above-mentioned properties only in $D>2m$.

Analogous to the Schwarzschild black hole in GR, the generic $N$-th order
Lovelock gravity admits static black hole solutions with spherical, planar and
hyperbolic horizons ($k=1,0,-1$) Boulware:1985wk ; Wheeler:1985nh ;
Wheeler:1985qd ; Myers:1988ze ; Cai:2003kt . Remarkably, as later shown in
o:2017sui , the fact that it admits such solutions satisfying
$g_{tt}g_{rr}=-1$ is a consequence of that the theory has a unitary massless
spin-2 graviton if the solution is to represent the exterior field of a
spherically symmetric distribution of mass.

With or without the cosmological constant, interesting features of the
thermodynamics of these solutions were discovered Myers:1988ze ; Clunan:2004tb
; Cai:2001dz ; Cai:2003gr ; Cai:2003kt ; Cvetic:2001bk . Although the equation
for the metric function, which is an $N$-th order polynomial equation, and the
resulting horizon structure is quite complicated, it is possible to study the
thermodynamics of these solutions with relative ease by finding the relevant
quantities in terms of the horizon radius Cai:2003kt  and see the effect of
the Lovelock terms on the thermodynamics without the explicit form of the
metric function. For example, one can show the universality of the
thermodynamics of Lovelock branes ($k=0$). Although the solutions differ,
their thermodynamics is identical to that of the static solution of GR. (see
Cadoni:2016hhd ; Hennigar:2017umz  for recent investigations).

The study of the thermodynamics of static black holes in Lovelock gravity also
led to an extended 1st law Kastor:2010gq , where the variations of the
couplings are included. When the cosmological constant is introduced as the
zeroth-order Lovelock term ($\mathcal{L}_{0}=1$), the mass of the black hole
is a function of the entropy $S$ and the coupling constants $c_{n\neq 1}$,
i.e., $M=M(S,c_{n\neq 1})$. Since $c_{n\neq 1}$ have the length unit
$\ell^{2(n-1)}$, using the Euler’s theorem suggest the following Smarr
relation

$(D-3)M=(D-2)TS+\sum_{n\neq 1}2(n-1)\mathcal{A}_{n}c_{n},$ (3)

with potentials $\mathcal{A}_{n}=\partialderivative{M}{c_{n}}$, and the
following extended form of the 1st law222Throughout the paper, we derive the
Smarr relation and the extended first law using the dimensionless form of the
coupling constants since the equation satisfied by the metric function takes a
simpler form with them.

$\delta M=T\delta S+\sum_{n\neq 1}\mathcal{A}_{n}\delta c_{n}.$ (4)

In Kastor:2010gq , generalizing the findings of Kastor:2009wy  for GR with a
cosmological constant, the Smarr relation (3) is verified geometrically from
Komar integrals for the Lovelock Lagrangians Kastor:2008xb  and the extended
1st law (4) was obtained through the Hamiltonian techniques. In this picture,
a negative cosmological constant can be seen as the thermodynamics pressure
and one obtains the notion of a thermodynamics volume
($\mathcal{A}_{0}c_{0}\sim VP$), allowing the interpretation of the black hole
mass $M$ as the enthalpy of the spacetime Kastor:2009wy . In this extended
phase space, novel phase transitions arise naturally (see Kubiznak:2016qmn
for a review of developments in this line of research called the black hole
chemistry.)

Recently, it was discovered that lower-dimensional ($D\leq 2m$) versions of
Lovelock gravity exist as scalar-tensor theories of Horndeski type
Horndeski:1974wa ; Kobayashi:2019hrl  with second-order field equations for
the metric and the scalar field, which were obtained by two different methods
that yield the same result under certain conditions. In the so-called Weyl
trick, one evaluates the difference of the $m$-th order Lovelock Lagrangian
for two conformally related metrics and finds a well-defined scalar-tensor
theory after a regularization of the coupling constant $c_{m}$. Alternatively,
one can perform a Kaluza-Klein reduction by again regularizing the relevant
coupling constant with an internal space that is conformally related to a
maximally symmetric space via the scalar $\phi$. In the latter approach, one
obtains terms that are proportional to curvature of the maximally symmetric
space. When the curvature is set to zero, the result agrees with that of the
former method. Throughout this work, we will refer to these scalar-tensor
theories as lower-dimensional Lovelock gravity. The 2nd- Fernandes:2020nbq ;
Lu:2020iav ; Kobayashi:2020wqy ; Hennigar:2020lsl  and 3rd-order Alkac:2022fuc
Lagrangians are as follows:

$\displaystyle\mathcal{L}^{\text{ST}}_{2}=$
$\displaystyle\phi\mathcal{L}_{2}+4G^{\mu\nu}\phi_{\mu}\phi_{\nu}-4X\square\phi+2X^{2},$
(5) $\displaystyle\mathcal{L}^{\text{ST}}_{3}=$
$\displaystyle\phi\mathcal{L}_{3}-3\phi\mathcal{L}_{2}X-12R(\square\phi)^{2}-48R_{\mu}^{\
\alpha}R_{\nu\alpha}\phi^{\mu}\phi^{\nu}$
$\displaystyle+24R_{\mu\nu}R\phi^{\mu}\phi^{\nu}-48R^{\mu\alpha}R_{\nu\alpha\rho\mu}\phi^{\nu}\phi^{\rho}+24R_{\alpha}^{\
\sigma\mu\nu}R_{\beta\sigma\mu\nu}\phi^{\alpha}\phi^{\beta}$
$\displaystyle+6RX^{2}+24R\phi_{\mu\nu}\phi^{\mu}\phi^{\nu}+12R\phi_{\mu\nu}\phi^{\mu\nu}-96R_{\mu}^{\
\alpha}\phi^{\mu}\phi^{\nu}\phi_{\alpha\nu}$
$\displaystyle+48R^{\mu\nu}\square\phi\phi_{\mu\nu}+48R_{\mu\nu}\square\phi\phi^{\mu}\phi^{\nu}-48R_{\mu\nu}X\phi^{\mu}\phi^{\nu}$
$\displaystyle-48R^{\mu\nu}\phi_{\alpha\nu}\phi^{\alpha}_{\
\mu}-48R_{\alpha\sigma\beta\mu}\phi^{\alpha}\phi^{\beta}\phi^{\mu\sigma}-24R_{\alpha\sigma\beta\mu}\phi^{\beta\alpha}\phi^{\mu\sigma}$
$\displaystyle-16(\square\phi)^{3}+24X(\square\phi)^{2}+96\square\phi\phi_{\mu\nu}\phi^{\mu}\phi^{\nu}-24X^{3}$
$\displaystyle-144X\phi_{\mu\nu}\phi^{\mu}\phi^{\nu}-96\phi^{\mu}\phi^{\nu}\phi_{\alpha\nu}\phi^{\alpha}_{\
\mu}-32\phi^{\mu\nu}\phi_{\alpha\mu}\phi^{\alpha}_{\ \nu}$
$\displaystyle+48\square\phi\phi_{\mu\nu}\phi^{\mu\nu}-24X\phi_{\mu\nu}\phi^{\mu\nu},$
(6)

where $\phi_{\mu}\equiv\partial_{\mu}\phi$,
$\phi_{\mu\nu}\equiv\nabla_{\mu}\nabla_{\nu}\phi$ and
$X\equiv\partial_{\mu}\phi\partial^{\mu}\phi$. Note that the first one is
valid for $D\leq 4$ and the second one applies when $D\leq 6$.

There has been an ongoing investigation of the properties of the lower-
dimensional Lovelock gravity in the literature Ma:2020ufk ; Hennigar:2020fkv ;
Hennigar:2020drx ; Khodabakhshi:2022knu ; Mao:2022zrf ; Alkac:2022zda ;
Bakopoulos:2022gdv ; Guajardo:2023uix . In this work, we aim to give an
analysis of the thermodynamics of static black solutions using the Euclidean
methods Regge:1974zd ; Gibbons:1976ue , which, as will be seen, makes the
connection between higher-dimensional pure gravity theories and the lower-
dimensional scalar-tensor theories transparent. In this approach, one gets the
usual $\delta M=T\delta S$ form of the 1st law. However; after obtaining the
Smarr relation from Euler’s theorem and finding a modified version of it,
which we will call a Smarr-like relation, in $D=4,6$ where the entropy is not
a homogeneous function (see Subsections 3.1 and 3.3 for details), the extended
1st law for these theories can be derived easily.

Additionally, we will give a derivation of the Smarr relation and its
modification that we call a Smarr-like relation from the scale symmetry of the
reduced action that follows from the ansatz which can be employed to find the
solution consistently. Initiated in Banados:2005hm , this approach has been
used for studying planar black holes ($k=0$) in different models
Gonzalez:2009nn ; Correa:2013bza ; Bravo-Gaete:2014haa ; Hyun:2015tia ;
Ahn:2015uza ; Perez:2015kea ; Hyun:2017nkb ; Erices:2017nta ; Erices:2019onl ;
Bravo-Gaete:2021hza  and how the method should be modified in the existence of
terms breaking the scale symmetry was presented in Ahn:2015shg . This
modification provides an efficient way to obtain the holographically
renormalized action and was successfully applied to various AdS/CMT models in
Hyun:2016isn . In such a case, instead of a conserved Noether charge that
exists in the scale-symmetric case, one can define a charge function that
evolves in the radial direction. But still, by evaluating it on the horizon
and at infinity, the Smarr relation can be derived. We will show that in
higher dimensions, one finds the Noether charge and then can derive the Smarr
relation also for solutions with non-planar event horizons in addition to the
planar case. The lower-dimensional theories that we will study in this paper
exhibit an exceptional behavior that has not been observed in the literature
before: Despite the scale-symmetry breaking terms in the reduced action that
appear for nonplanar horizons ($k\neq 0$), the charge function is conserved
on-shell for the solutions under study and the Smarr(-like) relation follows
straightforwardly.

The outline of this paper is as follows: In Section 2, we derive the static
black hole solution of the cubic Lovelock gravity in higher dimensions, where
it exists as a pure gravity theory, from a reduced action and study the
thermodynamics of the solution. Then, we give a derivation of the Smarr
relation based on the scale symmetry of the reduced action. In Section 3, we
apply the same techniques to scalar-tensor theories that are lower-dimensional
versions of cubic Lovelock gravity and discuss the similarities and
differences compared to the higher-dimensional case. We end our paper with a
summary of our results in Section 4.

##  2 Cubic Lovelock gravity in higher dimensions

###  2.1 General relativity with a cosmological constant in $D=4$

In order to introduce the methods that we will use in a simple setup, we will
first present the analysis of static black hole solutions of GR with the
cosmological constant in $D=4$, whose action is given by

$I=\frac{1}{16\pi
G}\int\differential[4]{x}\left[\frac{6}{L^{2}}\zeta+R\right],$ (7)

which corresponds to $D=4$,
$(c_{0},c_{1})=\left(\dfrac{6\zeta}{L^{2}},1\right)$ and
$(\mathcal{L}_{0},\mathcal{L}_{1})=(1,R)$ in the general form (1). In this
paper, we will use the dimensionless form of the coupling constants in
deriving the static black hole solutions and $\zeta$ is the first example.
Instead of using the covariant form of the field equations, one can insert the
ansatz

$\differential{s}^{2}=-N^{2}(r)h(r)\differential{t}^{2}+\frac{\differential{r}^{2}}{h(r)}+r^{2}\differential\Sigma^{2}_{2,k},$
(8)

where

$h(r)=k+\frac{r^{2}}{L^{2}}f(r),$ (9)

and

$\differential\Sigma_{2,k}=\gamma_{ij}\differential x^{i}\differential
x^{j}=\frac{\differential x^{2}}{1-kx^{2}}+k^{2}\differential\theta^{2},\qquad
i,j=2,3$ (10)

which corresponds to 2-dimensional spherical, planar and hyperbolic horizons
($k=+1,0,-1$), into the action (7) to find the following reduced action

$\displaystyle I=$
$\displaystyle-\Sigma_{2}\int\differential{t}\int\differential{r}\frac{r}{L^{2}}\left[N^{\prime}\left(3r^{3}f^{\prime}+10r^{2}f+4kL^{2}\right)\right.\qquad$
$\displaystyle\left.+rN\left(r^{2}f^{\prime\prime}+8rf^{\prime}+12f-6\zeta\right)+2rN^{\prime\prime}\left(r^{2}f+kL^{2}\right)\right],$
(11)

where $\Sigma_{2}=\int\sqrt{\gamma}\,\differential^{2}x$ and primes denote the
derivative with respect to the coordinate $r$. By integrating by parts, the
reduced action can be simplified into

$I=2\Sigma_{2}\int\differential{t}\int\differential{r}N\derivative{r}\left[\frac{r^{3}}{L^{2}}\left(\zeta-f\right)\right].$
(12)

The equations for the functions $\left\\{N(r),f(r)\right\\}$ can be found from
the variation of the reduced action. $\functionalderivative{I}{N}=0$ gives the
following algebraic equation for the function $f(r)$

$\zeta-f=\frac{\omega}{r^{3}},\qquad\omega:\text{constant}.$ (13)

An important point is that one obtains this simple form thanks to the form of
the function $h(r)$ given in (9), which will also be useful when we study more
general cases. $\functionalderivative{I}{f}=0$ is satisfied for $N(r)=1$,
which determines the class of metrics that we study in this paper.

In the Euclidean approach to thermodynamics, we identify the partition
function for a thermodynamic ensemble with the Euclidean path integral
evaluated by the saddle-point approximation around the Euclidean continuation
of the classical solution Gibbons:1976ue . In our case, the Euclidean
continuation of the metric is as follows

$\differential{s}^{2}_{\text{E}}=N^{2}(r)h(r)\differential{\tau}^{2}+\frac{\differential{r}^{2}}{h(r)}+r^{2}\differential\Sigma^{2}_{2,k},$
(14)

where $\tau=it$ is the periodic Euclidean time and the Euclidean and
Lorentzian actions are related by $I_{\text{E}}=-iI$. The periodicity of the
Euclidean time is the inverse temperature of the solution which is determined
by avoiding a conical singularity at the horizon. For $N(r)=1$, it reads

$\beta=\frac{1}{T}=\frac{4\pi}{h^{\prime}(r_{+})},$ (15)

where $r_{+}$ is the location of the event horizon, which is found from the
condition $h(r_{+})=0$. In terms of the metric function $f(r)$, these
relations imply,

$\displaystyle f(r_{+})$ $\displaystyle=-k\frac{L^{2}}{r_{+}^{2}},$ (16)
$\displaystyle\beta$ $\displaystyle=\frac{4\pi
r_{+}L^{2}}{-2kL^{2}+r_{+}^{3}f^{\prime}(r_{+})},$ (17)

which follows from eqn. (9). Note that so far we have used only the relation
between functions $h(r)$ and $f(r)$ in deriving these relations. Therefore,
they will be valid when we study the generalization of the line element (8) to
higher dimensions.

For static solutions of 4D GR with a cosmological constant, one has to impose
the equation for the metric function $f(r)$ given in (13). Setting $r=r_{+}$
in this equation (and its derivative) and using the relations (16, 17), we
find the integration constant $\omega$ and the inverse temperature $\beta$ for
our solutions as follows

$\displaystyle\omega$
$\displaystyle=L^{3}\left[\zeta\frac{r_{+}^{3}}{L^{3}}+k\frac{r_{+}}{L}\right],$
(18) $\displaystyle\beta$ $\displaystyle=\frac{4\pi
r_{+}}{3\zeta\frac{r_{+}^{2}}{L^{2}}+k}.$ (19)

The Euclidean action reads

$I_{\text{E}}=-2\beta\Sigma_{2}\int_{r_{+}}^{\infty}\differential{r}N\derivative{r}\left[\frac{r^{3}}{L^{2}}\left(\zeta-f\right)\right]+\int_{r_{+}}^{\infty}\derivative{B}{r},$
(20)

where the $\beta$ factor arises from the integration over the periodic
Euclidean time and the boundary term $B$ is fixed by requiring that the
Euclidean action $I_{\text{E}}$ has an extremum on-shell, i,e, $\variation
I_{\text{E}}\evaluated{}_{\text{on-shell}}=0$. For the reduced action in (20),
one finds

$\variation{B}=-2\beta\Sigma_{2}\frac{Nr^{3}}{L^{2}}\variation{f}.$ (21)

In studying the thermodynamics of the solutions, the variations should be
taken with respect to the parameters characterizing the mass and the entropy
of the solutions. In our case, we will work with the event horizon radius
$r_{+}$. With $N(r)=1$, the variation of the metric function $f(r)$ can be
found from eqn. (13) to be

$\variation{f}=-\frac{\variation{\omega}}{r^{3}},$ (22)

which can be related to the changes in $r_{+}$ via eqn. (18) but we will
proceed with this form because of its compactness. The variation of the
boundary term (21) becomes

$\variation{B}=\frac{2\beta\Sigma_{2}}{L^{2}}\variation{\omega}.$ (23)

Investigating the on-shell variation of the Euclidean action (20), we find

$\variation{I_{\text{E}}}\evaluated{}_{\text{on-shell}}=\variation
B\evaluated{}_{\infty}-\variation B\evaluated{}_{r_{+}}=0.$ (24)

Interpreting the on-shell Euclidean action as free energy, we make the
following identifications

$\variation B\evaluated{}_{\infty}=\beta\variation M,\qquad\qquad\variation
B\evaluated{}_{r_{+}}=\variation S.$ (25)

Note that since the boundary term is fixed to guarantee the stationarity of
the Euclidean action $I_{\text{E}}$, the first law is automatically satisfied
($\beta\variation M=\variation S$). The infinitesimal variations of the mass
and the entropy follow from the identifications (25) as

$\displaystyle\variation M$
$\displaystyle=\frac{2\Sigma_{2}}{L^{2}}\variation\omega,$ (26)
$\displaystyle\variation S$
$\displaystyle=\frac{2\beta\Sigma_{2}}{L^{2}}\variation\omega.$ (27)

The mass of the black hole solutions can easily be found in terms of the
integration constant $\omega$ to be

$M=\frac{2\Sigma_{2}}{L^{2}}\omega=2\Sigma_{2}r_{+}\left[\zeta\frac{r_{+}^{2}}{L^{2}}+k\right],$
(28)

where we have used (18). The entropy can be calculated after using the
expressions for the inverse temperature $\beta$ (19) and the integration
constant $\omega$ (18) in terms of the event horizon radius $r_{+}$, and then
integrating (27). The result is the celebrated area law given by

$S=4\pi\Sigma_{2}r_{+}^{2}.$ (29)

Since we have computed $M=M(r_{+},\zeta)$ and $S(r_{+})$, using Euler’s
theorem we can write

$\displaystyle M$
$\displaystyle=r_{+}\partialderivative{M}{r_{+}}-2\zeta\partialderivative{M}{\zeta},$
(30) $\displaystyle 2S$ $\displaystyle=r_{+}\partialderivative{S}{r_{+}}.$
(31)

From the chain rule, we have
$\partialderivative{M}{r_{+}}=\frac{1}{\beta}\partialderivative{S}{r_{+}}$.
Solving for $\partialderivative{S}{r_{+}}$ in (31) and inserting the result in
(30), we find the Smarr relation as

$\beta
M=2S-2\beta\zeta\mathcal{A}_{0},\qquad\quad\mathcal{A}_{0}=2\Sigma_{2}\frac{r_{+}^{3}}{L^{2}}.$
(32)

Considering the changes in the dimensionless coupling constant $\zeta$, the
extended first law can be written as

$\beta\variation M=\variation S+\beta\mathcal{A}_{0}\variation\zeta.$ (33)

This is the result first obtained in Kastor:2009wy  where the mass is
interpreted as the enthalpy of the spacetime and
$\mathcal{A}_{0}\variation\zeta$ term in the extended first law (33) is the
$V\variation{P}$ term in our conventions. We will repeat the same procedure
when dealing with more complicated expressions for the mass $M$ and the
entropy $S$.

The Smarr relation (32) can also be derived from the global scaling symmetry
of the reduced action Banados:2005hm . If the reduced action enjoys a scaling
symmetry for a Lagrangian $L_{\text{red}}[\Phi_{A},\Phi^{\prime}_{A},r]$ which
is a function of the functions $\Phi_{A}=\left\\{N(r),f(r)\right\\}$, their
first derivatives
$\Phi^{\prime}_{A}=\left\\{N^{\prime}(r),f^{\prime}(r)\right\\}$ and the
coordinate $r$, the associated Noether charge is

$Q=\sum_{A}\partialderivative{L_{\text{red}}}{\Phi^{{}^{\prime}}_{A}}\variation\Phi_{A}-rL_{\text{red}},$
(34)

where under a global scaling of the coordinate $r$
($r\rightarrow\tilde{r}=\Lambda r$, $\Lambda$: constant), the scaling weights
of the functions are defined as

$\Phi_{A}\rightarrow{\tilde{\Phi}_{A}(\tilde{r})}=\Lambda^{-\Delta_{A}}\Phi_{A}(r),$
(35)

which gives rise to the infinitesimal transformations

$\variation\Phi_{A}=r\Phi_{A}^{\prime}+\Delta_{A}\Phi_{A}.$ (36)

Our reduced action (20) is scale invariant with the scaling weights
$(\Delta_{N},\Delta_{f})=(3,0)$, from which the Noether charge is found to be

$Q=\frac{6\beta\Sigma_{2}Nr^{3}}{L^{2}}(\zeta-f),$ (37)

which is obviously conserved on-shell due to the equation (13) satisfied by
the metric function $f(r)$ and can be written in terms of the integration
constant $\omega$ as

$Q=\frac{6\beta\Sigma_{2}}{L^{2}}\omega.$ (38)

Evaluating the Noether charge $Q$ at infinity and at the horizon, we obtain

$Q\evaluated{}_{\infty}=3\beta M,\qquad
Q\evaluated{}_{r_{+}}=3\left(2S-2\beta\zeta\mathcal{A}_{0}\right),$ (39)

which are the left- and the right-hand-side of the Smarr relation (32)
multiplied by 3. Hence, we have derived it from the Noether charge of the
global scaling symmetry of the reduced action (20).

###  2.2 Cubic Lovelock gravity in $D>6$

Now, we are ready to analyze the thermodynamics of static black hole solutions
of cubic Lovelock gravity in higher dimensions ($D>6$) where it exits as a
pure gravity theory. The action of the theory is as follows

$I=\int\differential[D]{x}\sqrt{-g}\left[\frac{(D-1)(D-2)}{L^{2}}\zeta+R+\alpha_{2}L^{2}\mathcal{L}_{2}+\alpha_{3}L^{4}\mathcal{L}_{3}\right].$
(40)

In the general form of the Lagrangian (1), we have chosen

$(c_{0},c_{1},c_{2},c_{3})=\left(\frac{(D-1)(D-2)\zeta}{L^{2}},1,\alpha_{2}L^{2},\alpha_{3}L^{4}\right),$
(41)

where $\zeta$, $\alpha_{2}$ and $\alpha_{3}$ are the dimensionless coupling
constants corresponding to the 0th- ($\mathcal{L}_{0}=1$), 2nd- and 3rd-order
Lovelock Lagrangians.

The generalization of our metric ansatz to generic $D$-dimensions takes the
following form

$\differential{s}^{2}_{D}=-N(r)h(r)\differential{t}^{2}+\frac{\differential{r}^{2}}{h(r)}+r^{2}\differential\Sigma^{2}_{D-2,k},$
(42)

where we choose the function $h(r)$ again as in (9). The metric of
$(D-2)$-dimensional constant curvature space of unit radius is given by

$d\Sigma^{2}_{D-2,k}=\gamma_{ij}\differential x^{i}\differential
x^{j}=\frac{\differential{x}^{2}}{1-kx^{2}}+x^{2}\differential\Omega^{2}_{D-3},\qquad
i,j=2,\ldots D-1$ (43)

with
$\differential\Omega^{2}_{D-3}=\differential\theta^{2}_{1}+\sin^{2}{\theta_{1}}\differential\theta^{2}+\dots+\sin^{2}\theta_{1}\dots\sin^{2}\theta_{D-4}\differential\theta^{2}_{D-3}$
being the metric on a $(D-3)$-dimensional unit sphere. By appropriate
identifications, it is possible to make any constant $(r,t)$ section of the
spacetime compact Mann:1997iz . Substituting this ansatz into the action and
integrating by parts, we arrive at the reduced action

$I=(D-2)\Sigma_{D-2}\int\differential{t}\int\differential{r}N\derivative{r}\left[\frac{r^{D-1}}{L^{2}}\left[\zeta-f+\tilde{\alpha}_{2}f^{2}-\tilde{\alpha}_{3}f^{3}\right]\right],$
(44)

where $\Sigma_{D-2}=\int\sqrt{\gamma}\,\differential^{D-2}x$ and we rewrite
the effect of the couplings in a compact form as

$\tilde{\alpha}_{2}=(D-3)(D-4)\alpha_{2},\qquad\tilde{\alpha}_{3}=(D-3)(D-4)(D-5)(D-6)\alpha_{3}.$
(45)

We explicitly see that the 2nd- and 3rd-order Lovelock Lagrangians give a
nontrivial contribution to the reduced action and the equation for the metric
function $f(r)$ when $D>4$ and $D>6$ respectively. As before,
$\functionalderivative{I}{f}=0$ is satisfied for $N(r)=1$. From
$\functionalderivative{I}{N}=0$, we again find an algebraic equation for the
metric function $f(r)$

$\zeta-f+\tilde{\alpha}_{2}f^{2}-\tilde{\alpha}_{3}f^{3}=\frac{\omega}{r^{D-1}},\qquad\omega:\text{constant}.$
(46)

Remember that the expressions for the metric function evaluated at the horizon
radius $f(r_{+})$ (16) and the inverse temperature (17) are still valid since
they just follow the definition of the inverse temperature (15) and the
relation (9) between the functions $h(r)$ and $f(r)$. Using them together with
the equation (46) satisfied by the metric function $f(r)$ in cubic Lovelock
gravity in higher dimensions, one finds the following expressions for the
integration constant $\omega$ and the inverse temperature $\beta$

$\displaystyle\omega$
$\displaystyle=L^{2}r_{+}^{D-3}\left[\zeta\frac{r_{+}^{2}}{L^{2}}+k+\tilde{\alpha}_{2}k^{2}\frac{L^{2}}{r_{+}^{2}}+\tilde{\alpha}_{3}k^{3}\frac{L^{4}}{r_{+}^{4}}\right],$
(47) $\displaystyle\beta$ $\displaystyle=4\pi
r_{+}\left[\frac{1+2\tilde{\alpha}_{2}k\frac{L^{2}}{r_{+}^{2}}+3\tilde{\alpha}_{3}k^{2}\frac{L^{4}}{r_{+}^{4}}}{(D-1)\zeta\frac{r_{+}^{2}}{L^{2}}+(D-3)k+(D-5)\tilde{\alpha}_{2}k^{2}\frac{L^{2}}{r_{+}^{2}}+(D-7)\tilde{\alpha}_{3}k^{3}\frac{L^{4}}{r_{+}^{4}}}\right].$
(48)

From the Euclidean action

$I_{\text{E}}=-(D-2)\beta\Sigma_{D-2}\int_{r_{+}}^{\infty}\differential{r}N\derivative{r}\left[\frac{r^{D-1}}{L^{2}}\left[\zeta-f+\tilde{\alpha}_{2}f^{2}-\tilde{\alpha}_{3}f^{3}\right]\right]+\int_{r_{+}}^{\infty}\derivative{B}{r},$
(49)

one can easily show that in order to ensure $\variation
I_{\text{E}}\evaluated{}_{\text{on-shell}}=0$, the variation of the boundary
term should be taken as

$\delta
B=-(D-2)\beta\Sigma_{D-2}\frac{Nr^{D-1}}{L^{2}}\left[1-2\tilde{\alpha}_{2}f+3\tilde{\alpha}_{3}f^{2}\right]\delta
f.$ (50)

Taking the variation of the eqn. (46) and using the result (together with
$N(r)=1$) in the variation of the boundary term (50) give the following simple
expression

$\delta B=\frac{(D-2)\beta\Sigma_{D-2}}{L^{2}}\delta\omega.$ (51)

Employing the identifications (25), one obtains

$\displaystyle\variation M$
$\displaystyle=\frac{(D-2)\Sigma_{D-2}}{L^{2}}\variation\omega,$ (52)
$\displaystyle\variation S$
$\displaystyle=\frac{(D-2)\beta\Sigma_{D-2}}{L^{2}}\variation\omega,$ (53)

integration of which yields the following expressions for the mass333Note that
in the expression for the mass $M$, we use the compact form of the couplings
introduced in (45). and the entropy

$\displaystyle M$
$\displaystyle=(D-2)\Sigma_{D-2}r_{+}^{D-3}\left[\zeta\frac{r_{+}^{2}}{L^{2}}+k+\frac{\tilde{\alpha}_{2}k^{2}L^{2}}{r_{+}^{2}}+\frac{\tilde{\alpha}_{3}k^{3}L^{4}}{r_{+}^{4}}\right],$
(54) $\displaystyle S$
$\displaystyle=4\pi\Sigma_{D-2}r_{+}^{D-2}\left[1+\frac{2(D-3)!\alpha_{2}kL^{2}}{(D-1)!r_{+}^{2}}+\frac{3(D-5)!\alpha_{3}k^{3}L^{4}}{(D-1)!r_{+}^{4}}\right],$
(55)

which agree with the results of Cai:2003kt . While the integration for the
mass $M$ is trivial, for the entropy $S$ one has to use the expressions for
the inverse temperature $\beta$ (48) and the integration constant $\omega$
(47) in eqn. (53). Note that in the critical dimensions where the $m$-th order
Lovelock Lagrangian $\mathcal{L}_{m}$ is a boundary term, the equation (46)
for the metric function $f(r)$ and the mass $M$ gets no contribution. However,
as explicitly seen in the eqn. (55), the entropy is modified. Also, in $D>6$
dimensions, while the equation (46) for the metric function $f(r)$ gets non-
trivial contributions from the 2nd- and 3rd-order Lovelock Lagrangians
$\mathcal{L}_{2,3}$, for a solution with a planar horizon, the expressions for
the mass and the entropy coincide with those of GR. This is the universality
of the thermodynamics of Lovelock branes discussed in the literature
Cadoni:2016hhd ; Hennigar:2017umz .

In order to obtain the Smarr relation, we first use Euler’s theorem with
$M=M(r_{+},\zeta,\tilde{\alpha}_{2},\tilde{\alpha}_{3})$ and
$S=S(r_{+},\tilde{\alpha}_{2},\tilde{\alpha}_{3})$ to write

$\displaystyle(D-3)M$
$\displaystyle=r_{+}\partialderivative{M}{r_{+}}-2\zeta\partialderivative{M}{\zeta}+2\tilde{\alpha}_{2}\partialderivative{M}{\tilde{\alpha}_{2}}+4\tilde{\alpha}_{3}\partialderivative{M}{\tilde{\alpha}_{3}},$
(56) $\displaystyle(D-2)S$
$\displaystyle=r_{+}\partialderivative{S}{r_{+}}+2\tilde{\alpha}_{2}\partialderivative{S}{\tilde{\alpha}_{2}}+4\tilde{\alpha}_{3}\partialderivative{S}{\tilde{\alpha}_{3}}.$
(57)

Using these relations together with
$\partialderivative{M}{r_{+}}=\frac{1}{\beta}\partialderivative{S}{r_{+}}$
yields the Smarr relation as follows

$(D-3)\beta
M=(D-2)S+2\beta\left(-\zeta\mathcal{A}_{0}+\tilde{\alpha}_{2}\mathcal{A}_{2}+2\tilde{\alpha}_{3}\mathcal{A}_{3}\right),$
(58)
$(D-3)M=(D-2)TS-2\zeta\mathcal{A}_{0}+2\tilde{\alpha}_{2}\mathcal{A}_{2}+4\tilde{\alpha}_{3}\mathcal{A}_{3},$
(59)

where

$\displaystyle\begin{aligned}
\mathcal{A}_{0}=(D-2)\frac{\Sigma_{D-2}r_{+}^{D-1}}{L^{2}},\end{aligned}$ (60)
$\displaystyle\begin{aligned}
\mathcal{A}_{2}=\frac{(D-2)\Sigma_{D-2}kL^{2}r_{+}^{D-1}}{(D-4)\left(r_{+}^{4}+2\tilde{\alpha}_{2}kL^{2}r_{+}^{2}+3\tilde{\alpha}_{3}k^{2}L^{4}\right)}&\Big{[}-2\zeta(D-1)\frac{r_{+}^{2}}{L^{2}}-(D-2)k\\\
&+2\tilde{\alpha}_{2}k^{2}\frac{L^{2}}{r_{+}^{2}}+\tilde{\alpha}_{3}(D+2)k^{3}\frac{L^{4}}{r_{+}^{4}}\Big{]},\end{aligned}$
(61) $\displaystyle\begin{aligned}
\mathcal{A}_{3}=-\frac{(D-2)\Sigma_{D-2}k^{2}L^{4}r_{+}^{D-3}}{(D-6)\left(r_{+}^{4}+2\tilde{\alpha}_{2}kL^{2}r_{+}^{2}+3\tilde{\alpha}_{3}k^{2}L^{4}\right)}&\Big{[}3\zeta(D-1)\frac{r_{+}^{2}}{L^{2}}+(2D-3)k\\\
&+\tilde{\alpha}_{2}(D-3)k^{2}\frac{L^{2}}{r_{+}^{2}}-3\tilde{\alpha}_{3}k^{3}\frac{L^{4}}{r_{+}^{4}}\Big{]}.\end{aligned}$
(62)

This leads to the following form of the extended first law

$\beta\variation M=\variation
S+\beta\left[\mathcal{A}_{0}\variation\zeta+\mathcal{A}_{2}\variation\tilde{\alpha}_{2}+\mathcal{A}_{3}\variation\tilde{\alpha}_{3}\right],$
(63)

which was first proven in Kastor:2010gq  for generic $N$-th order Lovelock
gravity.

The reduced action (49) again enjoys a scaling symmetry with the weights
$(\Delta_{N},\Delta_{f})=(D-1,0)$. Calculating the Noether charge from (34),
we find

$Q=(D-1)(D-2)\beta\Sigma_{D-2}\frac{Nr^{D-1}}{L^{2}}\left[\zeta-\tilde{\alpha}_{1}f(r)+\tilde{\alpha}_{2}f^{2}(r)-\tilde{\alpha}_{3}f^{3}(r)\right],$
(64)

which is conserved on-shell for $N(r)=1$ due to the equation (46) satisfied by
the metric function $f(r)$. It can also be written in terms of the integration
constant $\omega$ as follows

$Q=(D-1)(D-2)\beta\Sigma_{D-2}\frac{\omega}{L^{2}}.$ (65)

When evaluated at infinity and at the horizon, it gives

$\displaystyle Q\evaluated{}_{\infty}$ $\displaystyle=(D-1)\beta M,$ (66)
$\displaystyle Q\evaluated{}_{r_{+}}$
$\displaystyle=\frac{D-1}{D-3}\left[(D-2)S+2\beta\left(-\zeta\mathcal{A}_{0}+\tilde{\alpha}_{2}\mathcal{A}_{2}+2\tilde{\alpha}_{3}\mathcal{A}_{3}\right)\right],$
(67)

which provides a derivation of the Smarr relation given in (58). Note that in
the previous studies of Lovelock gravity Correa:2013bza ; Bravo-Gaete:2021hza
, only planar black holes ($k=0$) were considered. However; we see that the
scaling symmetry is present even for black holes with non-planar horizons.

##  3 Cubic Lovelock gravity in lower dimensions

In this section, we will study the thermodynamics of the static black hole
solutions of cubic Lovelock gravity in lower dimensions, which is a scalar-
tensor theory. A detailed study of the solutions satisfying $g_{tt}g_{rr}=-1$
that corresponds to $N(r)=1$ in our metric ansatz (42) was done in
Alkac:2022fuc  via the reduced action that is obtained by employing the ansatz
(42) and assuming a radial profile for the scalar field, i.e., $\phi=\phi(r)$.
It was found that apart from a notable exception in $D=4$, there exist scalar
configurations such that the metric function $f(r)$ satisfies the algebraic
equation (46) with different $\tilde{\alpha}_{i}$’s ($i=2,3$) that are given
in Table 1. Note that they coincide with the values obtained by scaling them
by a factor of $\frac{1}{D-d}$ where $d$ is the lower dimension in which we
obtain the solution and then setting $D=d$. In Glavan:2019inb , it was claimed
that this procedure defines the Lovelock gravity in lower dimensions, which
was later refuted Gurses:2020ofy ; Gurses:2020rxb ; Arrechea:2020evj ;
Arrechea:2020gjw ; Ai:2020peo ; Fernandes:2020nbq ; Mahapatra:2020rds ;
Bonifacio:2020vbk ; Aoki:2020lig ; Shu:2020cjw ; Hohmann:2020cor ; Cao:2021nng
. In our setting, we realize these configurations as solutions of a well-
defined scalar-tensor theory.

$D$ | $\tilde{\alpha}_{2}$ | $\tilde{\alpha}_{3}$
---|---|---
3 | $-\alpha_{2}$ | $-6\alpha_{3}$
4 | $\alpha_{2}$ | $2\alpha_{3}$
5 | $2\alpha_{2}$ | $-2\alpha_{3}$
6 | $6\alpha_{2}$ | $6\alpha_{3}$
Table 1: Values of $\tilde{\alpha}_{2}$ and $\tilde{\alpha}_{3}$ for various
dimensions in terms of original dimensionless coupling constants, $\alpha_{2}$
and $\alpha_{3}$

The scalar field configurations that give rise to these solutions (with non-
zero $\tilde{\alpha}_{2}$ and $\tilde{\alpha}_{3}$) are given by Alkac:2022fuc

$\phi(r)=\begin{cases}\log(r)\pm\int_{r_{+}}^{r}\differential{u}\frac{L}{u}\frac{|k|}{\sqrt{L^{2}+ku^{2}f(u)}},&\text{$D=5,6$}\\\
\log(r).&\text{$D=3,4$}\end{cases}$ (68)

In $D=4$, the solution exists only with a planar horizon ($k=0$). When the
cubic term is discarded ($\tilde{\alpha}_{3}=0$), then non-planar horizons
($k=+1,-1$) are also allowed with the scalar field given by Hennigar:2020lsl ;
Lu:2020iav

$\phi(r)=\log(r)\pm\int_{r_{+}}^{r}\differential{u}\frac{L}{u}\frac{|k|}{\sqrt{L^{2}+ku^{2}f(u)}},\qquad
D=4,\tilde{\alpha}_{3}=0.$ (69)

Thermodynamical properties of these hairy black hole solutions can be found
from the reduced action as we presented in the previous section. However; as
will be seen, the reduced action for the scalar-tensor theories described by
the Lagrangians (5,1) does not possess a scaling symmetry. As shown in
Ahn:2015shg , one can proceed by defining a charge function $Q(r)$. If we
write the Lagrangian of the reduced action $L_{\text{red}}$ as a sum of terms
with different scaling weights $L_{\text{red}}^{[\Delta]}$ as

$L_{\text{red}}=\sum_{\Delta}L_{\text{red}}^{[\Delta]},\qquad\qquad{\tilde{L}_{\text{red}}^{[\Delta]}}=\Lambda^{-\Delta}L_{\text{red}}^{[\Delta]},$
(70)

then the charge function $Q(r)$ is defined as

$Q=\sum_{A}\partialderivative{L_{\text{red}}}{\Phi^{{}^{\prime}}_{A}}\variation\Phi_{A}-rL_{\text{red}},\qquad\qquad\Phi_{A}=\left\\{N,f,\xi=\phi^{\prime}\right\\},$
(71)

and it satisfies

$\derivative{Q}{r}=\sum_{\Delta}\left(\Delta-1\right)L_{\text{red}}^{[\Delta]}{\evaluated{}_{\text{on-
shell}}}.$ (72)

Note that when there is no term with a scaling weight other than $1$, i.e.
$L_{\text{red}}^{[\Delta\neq 1]}=0$, we obtain the usual Noether charge $Q$.
When the right-hand-side of the eqn. (72) is non-zero, there is a radial
evolution of the charge function $Q(r)$, which can be handled by integrating
from the event horizon radius $r_{+}$ to infinity. We find

$Q\evaluated{}_{\infty}-Q\evaluated{}_{r_{+}}=\int_{r_{+}}^{\infty}\differential{r}\sum_{\Delta}\left(\Delta-1\right)L_{\text{red}}^{[\Delta]}{\evaluated{}_{\text{on-
shell}}}.$ (73)

By relating the charge function evaluated at infinity and at the horizon to
the mass and the entropy respectively, one can still derive the Smarr(-like)
relation. However; we will see that for the solutions with $N(r)=1$ and
$\phi(r)$ given in (68) and (69), although there are terms with a scaling
dimension different than $1$, they vanish on-shell and the charge function
defined in (71) is still conserved. To our knowledge, these are the first
examples of hairy black holes with this property. Using the conserved charge,
it is straightforward to derive the Smarr(-like) relation. After the
discussion of these general features, we can now move on to the details in
different dimensions.

###  3.1 $D=6$

In $D=6$, while 1st- and 2nd-order Lovelock Lagrangians $\mathcal{L}_{1}=R$
and $\mathcal{L}_{2}$ exist, the 3rd-order Lagrangian $\mathcal{L}_{3}$ is a
boundary term. Therefore, in order to have a nontrivial contribution to field
equations, it should be replaced by the 3-rd order scalar-tensor version
$\mathcal{L}^{\text{ST}}_{3}$ given in (1). The action is as follows

$I=\int\differential[6]{x}\sqrt{-g}\left[\frac{20}{L^{2}}\zeta+R+\alpha_{2}L^{2}\mathcal{L}_{2}+\alpha_{3}L^{4}\mathcal{L}^{\text{ST}}_{3}\right].$
(74)

For $N(r)=1$ and the scalar field given in (68), the equation satisfied by the
metric function $f(r)$ is the algebraic equation (46) with
$\tilde{\alpha}_{2,3}$ given in Table 1, i.e.,

$\zeta-f+6\alpha_{2}f^{2}-6\alpha_{3}f^{3}=\frac{w}{r^{5}},\qquad\omega=\text{constant}.$
(75)

In order to study the thermodynamics of this solution, using the metric ansatz
(42) in $D=6$ and assuming $\phi=\phi(r)$, we find the reduced action as

$I_{\text{E}}=\int_{r_{+}}^{\infty}\differential{r}L_{\text{red}}+\int_{r_{+}}^{\infty}\derivative{B}{r},\qquad\qquad
L_{\text{red}}=\sum_{\Delta}L_{\text{red}}^{[\Delta]},$ (76)

where the parts of the reduced Lagrangian with different scaling weights are
given by

$\displaystyle\begin{aligned}
L_{\text{red}}^{[1]}=\frac{4\beta\Sigma_{4}Nr^{4}}{L^{2}}\Big{[}&-6f^{2}\left[5\alpha_{2}-3\alpha_{3}r^{6}\xi^{5}f^{\prime}+15\alpha_{3}r^{5}\xi^{4}f^{\prime}-30\alpha_{3}r^{4}\xi^{3}f^{\prime}\right.\\\
&\left.+30\alpha_{3}r^{3}\xi^{2}f^{\prime}-15\alpha_{3}r^{2}\xi
f^{\prime}\right]+f\left(5-12\alpha_{2}rf^{\prime}\right)\\\
&+rf^{\prime}+6\alpha_{3}rf^{3}\left[r^{5}\xi^{6}+6r^{4}\xi^{5}+\xi^{2}\left(36r^{3}\xi^{\prime}-69r\right)\right.\\\
&\left.-6\xi\left(4r^{2}\xi^{\prime}-5\right)+\xi^{4}\left(6r^{5}\xi^{\prime}-39r^{3}\right)\right.\\\
&\left.+\xi^{3}\left(76r^{2}-24r^{4}\xi^{\prime}\right)+6r\xi^{\prime}\right]-5\zeta\Big{]},\end{aligned}$
(77) $\displaystyle\begin{aligned}
L_{\text{red}}^{[3]}=&72\beta\Sigma_{4}\alpha_{3}kr^{3}N(r\xi-1)^{2}f\Big{[}2r\xi\left(r^{2}\xi^{2}-3r\xi+2\right)f^{\prime}\\\
&+f\left(r^{3}\xi^{4}+r\xi^{2}\left(6r^{2}\xi^{\prime}-19\right)+\xi\left(8-12r^{2}\xi^{\prime}\right)+6r^{2}\xi^{3}+2r\xi^{\prime}\right)\Big{]},\end{aligned}$
$\displaystyle\begin{aligned}
L_{\text{red}}^{[5]}=&72\beta\Sigma_{4}\alpha_{3}k^{2}L^{2}r^{2}N\xi(r\xi-2)\Big{[}r\xi\left(r^{2}\xi^{2}-3r\xi+2\right)f^{\prime}\\\
&+f\left(r^{3}\xi^{4}+r\xi^{2}\left(6r^{2}\xi^{\prime}-13\right)+\xi\left(6-12r^{2}\xi^{\prime}\right)+4r^{2}\xi^{3}+4r\xi^{\prime}\right)\Big{]},\end{aligned}$
$\displaystyle\begin{aligned}
L_{\text{red}}^{[7]}=24\beta\Sigma_{4}\alpha_{3}k^{3}L^{4}rN\xi^{2}(r\xi-2)^{2}\left(6r\xi^{\prime}+r\xi^{2}+4\xi\right).\end{aligned}$

Note that we have defined a new variable $\xi=\phi^{\prime}$. Since the scalar
field $\phi$ appears in the reduced action only with 1st- and 2nd-derivatives,
we proceed with this variable such that the expression for the charge function
(71) is still valid. Otherwise, it has to be modified for a reduced Lagrangian
$L_{\text{red}}$ that depends on the 2nd derivatives of the fields. As we see,
for a general case where no choice for the function $N(r)$ is made, the global
scaling symmetry is broken due to the terms with scaling weight different than
1 that arise from the 3rd-order scalar-tensor Lagrangian
$\mathcal{L}^{\text{ST}}_{3}$.

The variation of the boundary term that follows from the condition
$\variation{I_{\text{E}}}\evaluated{}_{\text{on-shell}}=0$ reads

$\displaystyle\variation{B}=$
$\displaystyle\frac{144\beta\Sigma_{4}Nr^{2}}{L^{2}}\alpha_{3}(kL^{2}+r^{2}f)\Big{[}kL^{2}\xi(r\xi-2)+rf(r\xi-1)^{2}\Big{]}^{2}\variation{\xi}$
(78)
$\displaystyle+\frac{4\beta\Sigma_{4}Nr^{3}}{L^{2}}\Bigg{[}r^{2}-12\alpha_{2}r^{2}f+18\alpha_{3}\xi\Big{\\{}k^{2}L^{4}\xi(r\xi-2)^{2}(r\xi-1)$
$\displaystyle+2kL^{2}rf(r\xi-2)(r\xi-1)^{3}+r^{3}f^{2}\big{(}5-10r\xi+10r^{2}\xi^{2}-5r^{3}\xi^{3}+r^{4}\xi^{4}\big{)}\Big{\\}}\Bigg{]}\variation{f},$

which can be used to study the thermodynamics of a solution with $N(r)\neq 1$
if it exists. When we concentrate on the case $N(r)=1$ where the scalar field
$\phi$ satisfies $\eqref{phisolns}$ and the metric function $f(r)$ satisfies
(75), the variation of the boundary term becomes

$\variation
B=-\frac{4\beta\Sigma_{4}r^{5}}{L^{2}}\Big{[}1-12\alpha_{2}f+18\alpha_{3}f^{2}\Big{]}\variation
f.$ (79)

Note that there remains no contribution from the variation of the scalar field
(no terms with $\variation{\xi}$) and we just need to follow the same steps in
the higher-dimensional case. The results for the integration constant
$\omega$, the inverse temperature $\beta$, the mass $M$ and the entropy $S$
are as follows

$\displaystyle\omega$
$\displaystyle=L^{2}r_{+}^{3}\left[\zeta\frac{r_{+}^{2}}{L^{2}}+k+6\alpha_{2}k^{2}\frac{L^{2}}{r_{+}^{2}}+6\alpha_{3}k^{3}\frac{L^{4}}{r_{+}^{4}}\right],$
(80) $\displaystyle\beta$ $\displaystyle=4\pi
r_{+}\left[\frac{1+12\alpha_{2}k\frac{L^{2}}{r_{+}^{2}}+18\alpha_{3}k^{2}\frac{L^{4}}{r_{+}^{4}}}{5\zeta\frac{r_{+}^{2}}{L^{2}}+3k+6\alpha_{2}k^{2}\frac{L^{2}}{r_{+}^{2}}-6\alpha_{3}k^{3}\frac{L^{4}}{r_{+}^{4}}}\right],$
(81) $\displaystyle M$
$\displaystyle=4\Sigma_{4}r_{+}^{3}\Bigg{[}\zeta\frac{r_{+}^{2}}{L^{2}}+k+6\alpha_{2}k^{2}\frac{L^{2}}{r_{+}^{2}}+6\alpha_{3}k^{3}\frac{L^{4}}{r_{+}^{4}}\Bigg{]},$
(82) $\displaystyle S$
$\displaystyle=4\pi\Sigma_{4}r_{+}^{4}\Bigg{[}1+24\alpha_{2}k\frac{L^{2}}{r_{+}^{2}}+72\alpha_{3}k^{2}\frac{L^{4}}{r_{+}^{4}}\log(r_{+})\Bigg{]}.$
(83)

Having obtained the mass $M=M(r_{+},\zeta,\alpha_{2},\alpha_{3})$ and the
entropy $S=S(r_{+},\alpha_{2},\alpha_{3})$, one can attempt to use the Euler’s
theorem to obtain the Smarr relation as we did before. However; the entropy
$S$ in (83) is not a homogoneous function due to the $\log(r_{+})$ contibution
from the 3rd-order scalar-tensor Lagrangian $\mathcal{L}^{\text{ST}}_{3}$. By
a careful examination, one finds the following Smarr-like relation

$3\beta
M=4S+2\beta\Big{[}-\zeta\mathcal{A}_{0}+\alpha_{2}\mathcal{A}_{2}+2\alpha_{3}\bar{\mathcal{A}}_{3}\Big{]},$
(84)

where

$\displaystyle\mathcal{A}_{0}$
$\displaystyle=\frac{4\Sigma_{4}r_{+}^{5}}{L^{2}},$ (85)
$\displaystyle\mathcal{A}_{2}$
$\displaystyle=\frac{24\Sigma_{4}kL^{2}r_{+}^{5}}{r_{+}^{4}+12\alpha_{2}kL^{2}r_{+}^{2}+18\alpha_{3}k^{2}L^{4}}\Bigg{[}-5\zeta\frac{r_{+}^{2}}{L^{2}}-2k+6\alpha_{2}k^{2}\frac{L^{2}}{r_{+}^{2}}+24\alpha_{3}k^{3}\frac{L^{4}}{r_{+}^{4}}\Bigg{]},$
(86) $\displaystyle\bar{\mathcal{A}}_{3}$
$\displaystyle=\frac{6\Sigma_{4}k^{2}L^{4}r_{+}^{3}}{r_{+}^{4}+12\alpha_{2}kL^{2}r_{+}^{2}+18\alpha_{3}k^{2}L^{4}}\Bigg{[}15\zeta\frac{r_{+}^{2}}{L^{2}}\big{(}1-4\log(r_{+})\big{)}+k\big{(}13-36\log(r_{+})\big{)}$
$\displaystyle+6\alpha_{2}k^{2}\frac{L^{2}}{r_{+}^{2}}\big{(}11-12\log(r_{+})\big{)}+18\alpha_{3}k^{3}\frac{L^{4}}{r_{+}^{4}}\big{(}3+4\log(r_{+})\big{)}\Bigg{]}.$
(87)

While $\mathcal{A}_{0}$ and $\mathcal{A}_{2}$ are the potentials that appear
in the extended 1st law, $\bar{\mathcal{A}}_{3}$ is related to the potential
$\mathcal{A}_{3}$ by

$\bar{\mathcal{A}}_{3}=\mathcal{A}_{3}+\frac{72\Sigma_{4}k^{2}L^{4}\pi}{\beta},$
(88)

and the extended 1st law takes the following form

$\beta\variation M=\variation
S+\beta\left(\mathcal{A}_{0}\variation\zeta+\mathcal{A}_{2}\variation\alpha_{2}+\mathcal{A}_{3}\variation\alpha_{3}\right).$
(89)

Since we have a modified potential $\bar{\mathcal{A}}_{3}$ in (84) which is
related to the potential $\mathcal{A}_{3}$ in the extended 1st law (89), we
prefer to call it a Smarr-like relation.

For this solution, one can calculate the charge function from (71).
Remarkably, it leads to the following conserved charge

$Q=\frac{20\beta\Sigma_{4}Nr^{5}}{L^{2}}\Bigg{[}\zeta-f+6\alpha_{2}f^{2}-6\alpha_{3}f^{3}\Bigg{]},$
(90)

whose derivative vanish due to (75). We have a case where the charge function
$Q(r)$ does not evolve in the radial direction, and its value at infinity and
at the horizon can be written as

$Q\evaluated{}_{\infty}=5\beta M,\qquad
Q\evaluated{}_{r_{+}}=\frac{5}{3}\left[4S+2\beta\left(-\zeta\mathcal{A}_{0}+\alpha_{2}\mathcal{A}_{2}+2\alpha_{3}\bar{\mathcal{A}}_{3}\right)\right],$
(91)

which just implies our Smarr-like relation given in (84).

###  3.2 $D=5$

In $D=5$, the action we consider is

$I=\int\differential[5]{x}\sqrt{-g}\left[\frac{12}{L^{2}}\zeta+R+\alpha_{2}L^{2}\mathcal{L}_{2}+\alpha_{3}L^{4}\mathcal{L}^{\text{ST}}_{3}\right],$
(92)

where $\mathcal{L}_{3}$ in higher dimensions is replaced by the scalar-tensor
Lagrangian $\mathcal{L}^{\text{ST}}_{3}$ since it vanishes in $D=5$. In
$\mathcal{L}^{\text{ST}}_{3}$ given in (1), the first term vanishes and as a
result of that there will be no log-type contribution to the entropy as in
$D=6$. Hence, we will be able to obtain the Smarr relation in the standard
form.

We study the solution with $N(r)=1$ and the scalar field $\phi$ given in (68),
for which the metric function satisfies

$\zeta-f+2\alpha_{2}f^{2}+2\alpha_{3}f^{3}=\frac{\omega}{r^{4}},\qquad\omega=\text{constant}.$
(93)

The integration constant $\omega$ and the inverse temperature $\beta$ are

$\displaystyle\omega=$ $\displaystyle
L^{2}r_{+}^{2}\left[\zeta\frac{r_{+}^{2}}{L^{2}}+k+2\alpha_{2}k^{2}\frac{L^{2}}{r_{+}^{2}}-2\alpha_{3}k^{3}\frac{L^{4}}{r_{+}^{4}}\right],$
(94) $\displaystyle\beta=$ $\displaystyle 2\pi
r_{+}\left[\frac{1+4\alpha_{2}\frac{L^{2}}{r_{+}^{2}}-6\alpha_{3}\frac{L^{4}}{r_{+}^{4}}}{2\zeta\frac{r_{+}^{2}}{L^{2}}+k+2\alpha_{3}k^{3}\frac{L^{4}}{r_{+}^{4}}}\right].$
(95)

The Lagrangian for the reduced action $L_{\text{red}}$ can be decomposed with
respect to the scaling weights as follows

$\displaystyle\begin{aligned} L_{\text{red}}^{[1]}=\frac{3\Sigma_{3}\beta
r^{3}N}{L^{2}}\Big{[}&-2f^{2}\left[4\alpha_{2}-12\alpha_{3}r^{6}\xi^{5}f^{\prime}+45\alpha_{3}r^{5}\xi^{4}f^{\prime}-60\alpha_{3}r^{4}\xi^{3}f^{\prime}\right]\\\
&+30\alpha_{3}r^{3}\xi^{2}f^{\prime}+4f\left(1-\alpha_{2}rf^{\prime}\right)+rf^{\prime}\\\
&+8\alpha_{3}r^{2}\xi
f^{3}\big{[}r^{4}\xi^{5}+\xi^{2}\left(34r-18r^{3}\xi^{\prime}\right)\\\
&+6r^{3}\xi^{4}+3r^{2}\xi^{3}\left(2r^{2}\xi^{\prime}-9\right)+3\xi\left(6r^{2}\xi^{\prime}-5\right)-6r\xi^{\prime}\big{]}-4\zeta\Big{]},\end{aligned}$
(96) $\displaystyle\begin{aligned}
L_{\text{red}}^{[3]}=&36\Sigma_{3}\beta\alpha_{3}kr^{3}N\xi(r\xi-1)f\Big{[}r\xi\left(4r^{2}\xi^{2}-11r\xi+7\right)f^{\prime}\\\
&+2f\left[r^{3}\xi^{4}+3r\xi^{2}\left(2r^{2}\xi^{\prime}-5\right)+\xi\left(7-12r^{2}\xi^{\prime}\right)+5r^{2}\xi^{3}+4r\xi^{\prime}\right]\Big{]},\end{aligned}$
$\displaystyle\begin{aligned}
L_{\text{red}}^{[5]}=&18\Sigma_{3}\beta\alpha_{3}k^{2}L^{2}rN\xi\Big{[}r\xi\left(4r^{3}\xi^{3}-15r^{2}\xi^{2}+16r\xi-4\right)f^{\prime}\\\
&+4(r\xi-1)f\left(r^{3}\xi^{4}+2r\xi^{2}\left(3r^{2}\xi^{\prime}-5\right)+\xi\left(2-12r^{2}\xi^{\prime}\right)+3r^{2}\xi^{3}+2r\xi^{\prime}\right)\Big{]},\end{aligned}$
$\displaystyle\begin{aligned}
L_{\text{red}}^{[7]}=24\Sigma_{3}\beta\alpha_{3}k^{3}L^{4}N\xi^{2}(r\xi-2)\Big{[}\xi\left(6r^{2}\xi^{\prime}-2\right)+r^{2}\xi^{3}-6r\xi^{\prime}+2r\xi^{2}\Big{]}.\end{aligned}$

For a general solution, the variation of the boundary term should be taken as

$\displaystyle\variation{B}=$
$\displaystyle\frac{144\beta\Sigma_{3}Nr}{L^{2}}\alpha_{3}\xi(kL^{2}+r^{2}f)(r\xi-1)\Big{[}kL^{2}\xi(r\xi-2)+rf(r\xi-1)^{2}\Big{]}\variation{\xi}$
(97)
$\displaystyle+\frac{3\beta\Sigma_{3}Nr^{2}}{L^{2}}\Bigg{[}r^{3}-4\alpha_{2}fr^{3}+\alpha_{3}\Big{\\{}-24k^{2}L^{4}\xi^{2}-84kL^{2}r^{2}f\xi^{2}+96k^{2}L^{4}r\xi^{3}$
$\displaystyle+216kL^{2}r^{3}f\xi^{3}-90k^{2}L^{4}r^{2}\xi^{4}-180kL^{2}r^{4}\xi^{4}+24k^{2}L^{4}r^{3}\xi^{5}+48kL^{2}r^{5}f\xi^{5}$
$\displaystyle+6r^{4}f^{2}\xi^{2}(-10+20r\xi-15r^{2}\xi^{2}+4r^{3}\xi^{3})\Big{\\}}\Bigg{]}\variation{f},$

which again simplifies for our solution significantly and becomes

$\variation{B}=-\frac{3\beta\Sigma_{3}r^{4}}{L^{2}}\Bigg{[}1-4\alpha_{2}f-6\alpha_{3}f^{2}\Bigg{]}\variation{f}.$
(98)

From this, as explained before, it is straightforward to find the mass, the
entropy and the Smarr relation as

$\displaystyle M=$ $\displaystyle
3\Sigma_{3}r_{+}^{2}\left[\zeta\frac{r_{+}^{2}}{L^{2}}+k+2\alpha_{2}k^{2}\frac{L^{2}}{r_{+}^{2}}-2\alpha_{3}k^{3}\frac{L^{4}}{r_{+}^{4}}\right],$
(99) $\displaystyle S=$ $\displaystyle
4\pi\Sigma_{3}r_{+}^{3}\left[1+12\alpha_{2}k\frac{L^{2}}{r_{+}^{2}}+18\alpha_{3}k^{2}\frac{L^{4}}{r_{+}^{4}}\right],$
(100) $\displaystyle 2\beta M=$ $\displaystyle
3S+2\beta\Big{[}-\zeta\mathcal{A}_{0}+\alpha_{2}\mathcal{A}_{2}+2\alpha_{3}\mathcal{A}_{3}\Big{]},$
(101)

where the potentials are given by

$\displaystyle\mathcal{A}_{0}$
$\displaystyle=\frac{3\Sigma_{3}r_{+}^{4}}{L^{2}},$ (102)
$\displaystyle\mathcal{A}_{2}$
$\displaystyle=\frac{3\Sigma_{3}kL^{2}r_{+}^{4}}{r_{+}^{4}+4\alpha_{2}kL^{2}r_{+}^{2}-6\alpha_{3}k^{2}L^{4}}\Big{[}-8\zeta\frac{r_{+}^{2}}{L^{2}}-3k+4\alpha_{2}k^{2}\frac{L^{2}}{r_{+}^{2}}-14\alpha_{3}k^{3}\frac{L^{4}}{r_{+}^{4}}\Big{]},$
(103) $\displaystyle\mathcal{A}_{3}$
$\displaystyle=\frac{6\Sigma_{3}k^{2}L^{4}r_{+}^{4}}{r_{+}^{6}+4\alpha_{2}kL^{2}r_{+}^{4}-6\alpha_{3}k^{2}L^{4}r_{+}^{2}}\Big{[}12\zeta\frac{r_{+}^{2}}{L^{2}}+7k+4\alpha_{2}k^{2}\frac{L^{2}}{r_{+}^{2}}+6\alpha_{3}k^{3}\frac{L^{4}}{r_{+}^{4}}\Big{]},$
(104)

and the extended 1st-law (89) is satisfied with these potentials.

The expression for the charge function (71) leads to the following Noether
charge

$Q=\frac{12\Sigma_{3}\beta
r^{4}}{L^{2}}\Big{[}\zeta-f+2\alpha_{2}f^{2}+2\alpha_{3}f^{3}\Big{]},$ (105)

which is conserved due to (93) and gives a derivation of the Smarr relation
(101) with its values at infinity and the horizon given as

$Q\evaluated{}_{\infty}=4\beta M,\qquad
Q\evaluated{}_{r_{+}}=6S+4\beta\left(-\zeta\mathcal{A}_{0}+\alpha_{2}\mathcal{A}_{2}+2\alpha_{3}\mathcal{A}_{3}\right),$
(106)

###  3.3 $D=4$

In $D=4$, $\mathcal{L}_{3}$ vanishes and $\mathcal{L}_{2}$ is a boundary term.
Therefore, we replace them by the scalar-tensor Lagrangians
$\mathcal{L}^{\text{ST}}_{2,3}$ given in (5-1) and consider the action

$I=\int\differential[4]{x}\sqrt{-g}\left[\frac{6}{L^{2}}\zeta+R+\alpha_{2}L^{2}\mathcal{L}^{\text{ST}}_{2}+\alpha_{3}L^{4}\mathcal{L}^{\text{ST}}_{3}\right].$
(107)

As mentioned at the beginning of this section, black hole solutions with non-
planar horizons exist only when $\alpha_{3}=0$. Before we distinguish
different cases, it is useful to study the reduced action and the boundary
term for a general solution described by the ansatz (42) in $D=5$ and
$\phi=\phi(r)$. The Lagrangian of the reduced action with different scaling
weights are

$\displaystyle\begin{aligned} L_{\text{red}}^{[1]}=\frac{2\Sigma_{2}\beta
r^{2}N}{L^{2}}\bigg{[}&rf^{2}\big{[}36\alpha_{3}r^{5}\xi^{5}f^{\prime}-r^{3}\xi^{4}\left(\alpha_{2}+90\alpha_{3}rf^{\prime}\right)\\\
&+4r^{2}\xi^{3}\left(15\alpha_{3}rf^{\prime}-\alpha_{2}\right)\big{]}+rf^{2}\big{[}2\alpha_{2}r\xi^{2}\left(7-2r^{2}\xi^{\prime}\right)\\\
&+4\alpha_{2}\xi\left(2r^{2}\xi^{\prime}-3\right)-4\alpha_{2}r\xi^{\prime}\big{]}+rf^{\prime}\\\
&+f\left[3-2\alpha_{2}r^{4}\xi^{3}f^{\prime}+6\alpha_{2}r^{3}\xi^{2}f^{\prime}-6\alpha_{2}r^{2}\xi
f^{\prime}\right]\\\
&+6\alpha_{3}r^{3}\xi^{2}f^{3}\left[2r^{3}\xi^{4}+3r\xi^{2}\left(4r^{2}\xi^{\prime}-11\right)-4\xi\left(6r^{2}\xi^{\prime}-5\right)\right]\\\
&+72\alpha_{3}f^{3}r^{4}\xi^{2}\left(r\xi^{3}+\xi^{\prime}\right)-3\zeta\bigg{]},\end{aligned}$
(108) $\displaystyle\begin{aligned} L_{\text{red}}^{[3]}=4\Sigma_{2}\beta
rkN\Big{[}&-\alpha_{2}r\xi
f^{\prime}\left(r^{2}\xi^{2}-3r\xi+2\right)-36\alpha_{3}r^{5}\xi^{5}ff^{\prime}\\\
&+2r^{2}\xi^{3}f\left(\alpha_{2}-27\alpha_{3}rf^{\prime}\right)+\alpha_{2}r\xi^{2}f\left(4r^{2}\xi^{\prime}-9\right)+2\alpha_{2}r\xi^{\prime}f\\\
&+\xi
f\left(4\alpha_{2}-8\alpha_{2}r^{2}\xi^{\prime}\right)+6f^{2}r^{3}\alpha_{3}\xi^{4}\left(-35+3r^{2}\xi^{2}+18r^{2}\xi^{\prime}\right)\\\
&+12f^{2}r^{2}\alpha_{3}\xi^{2}\left[9\xi+6r^{2}\xi^{3}+2r(4-9r\xi)\xi^{\prime}\right]\\\
&+r^{3}\xi^{4}f\left(\alpha_{2}+90\alpha_{3}rf^{\prime}\right)\Big{]},\end{aligned}$
$\displaystyle\begin{aligned} L_{\text{red}}^{[5]}=2\Sigma_{2}\beta
L^{2}k^{2}N\xi\Big{[}&36\alpha_{3}r^{3}\xi^{4}\left(rf^{\prime}+2f\right)-48\alpha_{3}r\xi^{2}\left[f\left(9r^{2}\xi^{\prime}-2\right)-rf^{\prime}\right]\\\
&+r^{2}\xi^{3}\left(-\alpha_{2}-90\alpha_{3}rf^{\prime}+6\alpha_{3}f\left(36r^{2}\xi^{\prime}-41\right)\right)\\\
&+36\alpha_{3}r^{4}\xi^{5}f+4\xi\left(\alpha_{2}-r^{2}\xi^{\prime}(\alpha_{2}-42\alpha_{3}f)\right)\\\
&+8\alpha_{2}r\xi^{\prime}\Big{]},\end{aligned}$ $\displaystyle\begin{aligned}
L_{\text{red}}^{[7]}=&24\Sigma_{2}\beta\alpha_{3}k^{3}L^{4}N\xi^{2}\Big{[}\xi^{2}\left(6r^{2}\xi^{\prime}-2\right)+r^{2}\xi^{4}-12r\xi\xi^{\prime}+4\xi^{\prime}\Big{]}.\end{aligned}$

As we see, terms breaking the scaling symmetry arise from the scalar-tensor
Lagrangians $\mathcal{L}^{\text{ST}}_{m=2,3}$ (when $k\neq 0$). The variation
of the boundary term should be given by

$\displaystyle\variation{B}=-$
$\displaystyle\frac{2\beta\Sigma_{2}N}{L^{2}}\bigg{[}\alpha_{2}\Big{\\{}-4r^{2}f(kL^{2}+r^{2}f)+8kL^{2}r(kL^{2}+r^{2}f)\xi$
(109)
$\displaystyle+8r^{3}f(kL^{2}+r^{2}f)\xi-4kL^{2}r^{2}(kL^{2}+r^{2}f)\xi^{2}-4r^{4}f(kL^{2}+r^{2}f)\xi^{2}\Big{\\}}$
$\displaystyle+\alpha_{3}\Big{\\{}48k^{2}L^{4}(kL^{2}+r^{2}f)\xi^{2}+120kL^{2}r^{2}f(kL^{2}+r^{2}f)\xi^{2}$
$\displaystyle-144k^{2}L^{4}r(kL^{2}+r^{2}f)\xi^{3}-288kL^{2}r^{3}f(kL^{2}+r^{2}f)\xi^{3}$
$\displaystyle+72k^{2}L^{4}r^{2}(kL^{2}+r^{2}f)\xi^{4}+144kL^{2}r^{4}f(kL^{2}+r^{2}f)\xi^{4}$
$\displaystyle+72r^{4}f^{2}(kL^{2}+r^{2}f)\xi^{2}(r\xi-1)^{2}\Big{\\}}\bigg{]}\variation{\xi}$
$\displaystyle-\frac{2N\Sigma_{2}\beta}{L^{2}}\bigg{[}r^{3}+\alpha_{2}\Big{\\{}-2r^{2}(2kL^{2}+3r^{2}f)\xi+6r^{3}(kL^{2}+r^{2}f)\xi^{2}$
$\displaystyle-2r^{4}(kL^{2}+r^{2}f)\xi^{3}\Big{\\}}+\alpha_{3}\Big{\\{}48kL^{2}r^{2}(kL^{2}+r^{2}f)\xi^{3}+60r^{2}f(kL^{2}+r^{2}f)\xi^{3}$
$\displaystyle-90r^{3}(kL^{2}+r^{2}f)\xi^{4}+36r^{2}(kL^{2}r+r^{3}f)^{2}\xi^{5}\Big{\\}}\bigg{]}\variation{f},$

which can be used to study the thermodynamics of a solution.

####  3.3.1 $D=4$, $\alpha_{3}\neq 0$, $k=0$

When $\alpha_{3}\neq 0$, only a black hole with a planar horizon ($k=0$) is
allowed with the scalar field given in (68) and the metric function $f(r)$
satisfies

$\zeta-f+\alpha_{2}f^{2}-2\alpha_{3}f^{3}=\frac{\omega}{r^{3}},\qquad\omega=\text{constant}.$
(110)

The integration constant $\omega$ and the inverse temperature $\beta$ take
particularly simple forms as follows

$\omega=\zeta r_{+}^{3},\qquad\qquad\beta=\frac{4\pi L^{2}}{3\zeta r_{+}}.$
(111)

Together with the variation of the boundary term

$\variation
B=-\frac{2\beta\Sigma_{2}r^{3}}{L^{2}}\Big{[}1-2\alpha_{2}f+6\alpha_{3}f^{2}\Big{]},$
(112)

which simplifies into $\variation B=8\pi\Sigma_{2}r_{+}\variation r_{+}$, they
yield the thermodynamical relations

$\displaystyle M$
$\displaystyle=2\zeta\Sigma_{2}\frac{r_{+}^{3}}{L^{2}},\qquad
S=4\pi\Sigma_{2}r_{+}^{2},\qquad\beta M=2S-2\beta\zeta\mathcal{A}_{0}$ (113)
$\displaystyle\beta\variation{M}$
$\displaystyle=\variation{S}+\beta\mathcal{A}_{0}\variation{\zeta},\qquad\mathcal{A}_{0}=\frac{2\Sigma_{2}r_{+}^{3}}{L^{2}}.$
(114)

The Smarr relation in (113) can be easily derived from the Noether charge

$Q=\frac{6\Sigma_{2}\beta
r^{3}}{L^{2}}\left[\zeta-f+\alpha_{2}f^{2}-2\alpha_{3}f^{3}\right],$ (115)

which is conserved from (110), and whose values at infinity and the horizon
are given as

$Q\evaluated{}_{\infty}=3\beta M,\qquad
Q\evaluated{}_{r_{+}}=6S-6\beta\zeta\mathcal{A}_{0}.$ (116)

####  3.3.2 $D=4$, $\alpha_{3}=0$, $k=+1,0,-1$

For $\alpha_{3}=0$, non-planar horizons are also admitted provided that the
scalar field is given by (69). For this case, with the addition of the
conformal coupling of the scalar, the mass and the entropy were computed in
Babichev:2022awg  by Euclidean methods. Here, we reproduce them to discuss the
scaling properties, Smarr relation and the extended first law.

The metric function satisfies the eqn. (110) with $\alpha_{3}=0$. The
integration constant $\omega$ and the inverse temperature $\beta$ are more
complicated due to the $k$-dependent terms as follows

$\displaystyle\omega$
$\displaystyle=L^{2}r_{+}\left[\zeta\frac{r_{+}^{2}}{L^{2}}+k+\alpha_{2}k^{2}\frac{L^{2}}{r_{+}^{2}}\right],$
(117) $\displaystyle\beta$ $\displaystyle=4\pi
r_{+}\left[\frac{1+2\alpha_{2}k\frac{L^{2}}{r_{+}^{2}}}{3\zeta\frac{r_{+}^{2}}{L^{2}}+k-\alpha_{2}k^{2}\frac{L^{2}}{r_{+}^{2}}}\right].$
(118)

For this solution, although it follows from a different scalar field
configuration, the variation of the boundary term is given by (112) with
$\alpha_{3}=0$. Using this, the thermodynamics of the solution follows
straightforwardly:

$\displaystyle M$
$\displaystyle=2\Sigma_{2}r_{+}\left[\zeta\frac{r_{+}^{2}}{L^{2}}+k+2\alpha_{2}k^{2}\frac{L^{2}}{r_{+}^{2}}\right],\qquad
S=4\pi\Sigma_{2}r_{+}^{2}\left[1+4\alpha_{2}k\frac{L^{2}}{r_{+}^{2}}\log(r_{+})\right],$
(119) $\displaystyle\beta M$
$\displaystyle=2S+2\beta\left[-\zeta\mathcal{A}_{0}+\alpha_{2}\mathcal{\bar{A}}_{2}\right],\qquad\beta\variation{M}=\variation{S}+\beta\left(\mathcal{A}_{0}\variation{\zeta}+\mathcal{A}_{2}\variation{\alpha_{2}}\right),$
(120) $\displaystyle\mathcal{A}_{0}$
$\displaystyle=\frac{2\Sigma_{2}r_{+}^{3}}{L^{2}},\qquad\mathcal{A}_{2}=\frac{2\Sigma_{2}kL^{2}}{r_{+}}\left[k-\frac{2(3\zeta
r_{+}^{4}+kL^{2}-\alpha_{2}k^{2}L^{4})\log(r_{+})}{r_{+}^{2}+2\alpha_{2}kL^{2}}\right],$
(121) $\displaystyle\mathcal{\bar{A}}_{2}$
$\displaystyle=\mathcal{A}_{2}+\frac{8\pi}{\beta}\Sigma_{2}kL^{2}.$ (122)

As in $D=6$ where $\mathcal{L}_{3}$ is a boundary term, the entropy in (119)
is not a homogeneous function and in the Smarr-like relation in (120) we have
a modification to the potential $\mathcal{A}_{2}$ that appear in the extended
1st law in (120). The Noether charge is given by the expression in (115) with
$\alpha_{3}=0$ and produces the Smarr-like relation (119) with its values at
the event horizon radius $r_{+}$ and infinity, which are given by

$Q\evaluated{}_{\infty}=3\beta M,\qquad
Q\evaluated{}_{r_{+}}=6S+6\beta\left[-\zeta\mathcal{A}_{0}+\alpha_{2}\bar{\mathcal{A}}_{2}\right].$
(123)

###  3.4 $D=3$

Since both $\mathcal{L}_{2,3}$ vanish in $D=3$, we consider the following
action

$I=\int\differential[3]{x}\sqrt{-g}\left[\frac{2}{L^{2}}\zeta+R+\alpha_{2}L^{2}\mathcal{L}^{\text{ST}}_{2}+\alpha_{3}L^{4}\mathcal{L}^{\text{ST}}_{3}\right],$
(124)

where the first term in (1) vanishes.

$D=3$ is a special case for two reasons. First, since the event horizon is
one-dimensional, it cannot have any curvature. Therefore, one should take
$k=0$ in the metric ansatz (42). Second, in addition to the solution (46)
discussed at the beginning of this section, the static Banados-Teitelboim-
Zanelli (BTZ) black hole Banados:1992wn  is also a solution.

We can start by considering the general form of the Lagrangian of the reduced
action and the variation of the boundary term that is required to study the
thermodynamics of a solution. They are given by

$\displaystyle L_{\text{red}}^{[1]}$ $\displaystyle=\frac{\Sigma_{1}\beta
r}{L^{2}}\Big{[}2f\left(1-2\alpha_{2}r^{4}\xi^{3}f^{\prime}+3\alpha_{2}r^{3}\xi^{2}f^{\prime}\right)+rf^{\prime}-8\alpha_{2}r^{3}f^{2}\xi\left(\xi^{2}-\xi^{\prime}\right)$
$\displaystyle+2r^{2}f^{2}\xi\left[36\alpha_{3}r^{4}\xi^{4}f^{\prime}-r^{2}\xi^{3}\left(\alpha_{2}+45\alpha_{3}rf^{\prime}\right)+\xi\left(6\alpha_{2}-4\alpha_{2}r^{2}\xi^{\prime}\right)\right]$
(125)
$\displaystyle+12\alpha_{3}r^{4}f^{3}\xi^{3}\left[3\xi\left(4r^{2}\xi^{\prime}-5\right)+2r^{2}\xi^{3}-12r\xi^{\prime}+12r\xi^{2}\right]-2\zeta\Big{]},$
$\displaystyle\variation B$
$\displaystyle=-\frac{\beta\Sigma_{1}N}{L^{2}}\bigg{[}r^{2}\Big{\\{}1+2\alpha_{2}r^{2}f(3-2r\xi)\xi^{2}+18\alpha_{3}r^{4}f^{2}(4r\xi-5)\xi^{4}\Big{\\}}\variation
f$
$\displaystyle\qquad\qquad\quad\,\,+8r^{4}f^{2}\xi(r\xi-1)(-\alpha_{2}+18\alpha_{3}r^{2}f\xi^{2})\variation\xi\bigg{]}.$
(126)

Note that terms that break the scaling symmetry $L_{\text{red}}^{[\Delta\neq
1]}$ in $D=6,5,4$ always come with a $k$ factor. In $D=3$, we see that,
despite the scalar hair, the Lagrangian of the reduced action has no symmetry-
breaking terms. Since we need to take $k=0$ to derive the solutions, this is
completely parallel to these higher dimensional cases where the event horizon
might have curvature.

####  3.4.1 Solution that satisfies the polynomial equation

For the scalar field given in (68), the metric function $f(r)$ satisfies the
following polynomial equation

$\zeta-f-\alpha_{2}f^{2}+6\alpha_{3}f^{3}=\frac{\omega}{r^{2}},\qquad\omega=\text{constant}.$
(127)

From this equation and the variation of the boundary term (126), the
thermodynamics of the solution can be obtained as follows:

$\displaystyle\omega$ $\displaystyle=\zeta
r_{+}^{2},\qquad\qquad\beta=\frac{2\pi L^{2}}{\zeta r_{+}},$ (128)
$\displaystyle\variation B$
$\displaystyle=-\frac{\beta\Sigma_{1}r^{2}}{L^{2}}\left(1+2\alpha_{2}f-18\alpha_{3}f^{2}\right)\variation
f,$ (129) $\displaystyle M$
$\displaystyle=\frac{\zeta\Sigma_{1}r_{+}^{2}}{L^{2}},\qquad\qquad
S=4\pi\Sigma_{1}r_{+},$ (130) $\displaystyle 0$
$\displaystyle=S+2\beta\left[-\zeta\mathcal{A}_{0}+\alpha_{2}\mathcal{A}_{2}+2\alpha_{3}\mathcal{A}_{3}\right],$
(131) $\displaystyle\mathcal{A}_{0}$
$\displaystyle=\frac{\Sigma_{1}r_{+}^{2}}{L^{2}},\qquad\mathcal{A}_{2}=0,\qquad\mathcal{A}_{3}=0,$
(132) $\displaystyle\beta\variation M$ $\displaystyle=\variation
S+\beta\left(\mathcal{A}_{0}\variation\zeta+\mathcal{A}_{2}\variation\alpha_{2}+\mathcal{A}_{3}\variation\alpha_{3}\right).$
(133)

The Smarr relation (131) has an alternative form in $D=3$ that is given by

$\beta M=\frac{1}{2}S,$ (134)

and this form of it can be derived from the Noether charge. For this solution,
it reads

$Q=\frac{2\beta\Sigma_{1}r^{2}}{L^{2}}\left[\zeta-f-\alpha_{2}f^{2}+6\alpha_{3}f^{3}\right],$
(135)

whose conservation is obvious from (127). Its values at infinity and at the
horizon are

$Q\evaluated{}_{\infty}=2\beta M,\qquad
Q\evaluated{}_{r_{+}}=S=2\beta\zeta\mathcal{A}_{0}.$ (136)

####  3.4.2 Static BTZ black hole

The static BTZ black hole with the line element

$\differential{s}^{2}=-\left[\frac{r^{2}}{\ell^{2}}-m\right]\differential{t}^{2}+\frac{1}{\dfrac{r^{2}}{\ell^{2}}-m}\differential{r}^{2}+r^{2}\differential{\varphi}^{2},$
(137)

where the event horizon is located at $r_{+}=\sqrt{m}\ell$, is a solution of
the theory described by the action (124) provided that the scalar field is
given by Alkac:2022fuc

$\phi(r)=\sqrt{\frac{\alpha_{2}}{18\alpha_{3}}}\int_{r_{+}}^{r}\frac{\differential{u}}{u\sqrt{f(u)}},$
(138)

for which the metric function $f(r)$ becomes

$f(r)=\frac{\alpha_{2}^{3}+972\zeta\alpha_{3}^{2}}{54\alpha_{2}^{2}\alpha_{3}+972\alpha_{3}}-\frac{mL^{2}}{r^{2}},$
(139)

which corresponds to the line element (137) with

$\frac{\ell^{2}}{L^{2}}=\frac{54\alpha_{2}^{2}\alpha_{3}+972\alpha_{3}^{2}}{\alpha_{2}^{3}+972\zeta\alpha_{3}^{2}}.$
(140)

Similar to our previous computations, we can write the metric function $f(r)$
in terms of an integration constant $\omega$ as

$\zeta+\frac{\alpha_{2}^{3}}{972\alpha_{3}^{2}}-\left[1+\frac{\alpha_{2}^{2}}{18\alpha_{3}}\right]f=\frac{\omega}{r^{2}},\qquad\omega=r_{+}^{2}\left[\zeta+\frac{\alpha_{2}^{3}}{972\alpha_{3}^{2}}\right]=\text{constant}.$
(141)

The inverse temperature $\beta$ in this case takes the following form

$\beta=\frac{2\pi
L^{2}}{r_{+}}\left[\frac{54\alpha_{2}^{2}\alpha_{3}+972\alpha_{3}^{2}}{\alpha_{2}^{3}+972\zeta\alpha_{3}^{2}}\right].$
(142)

The variation of the boundary term (126) becomes

$\variation{B}=-\frac{\beta\Sigma_{1}r^{2}}{L^{2}}\Big{[}1+\frac{\alpha_{2}^{2}}{18\alpha_{3}}\Big{]}\variation{f}=\frac{\beta\Sigma_{1}}{L^{2}}\variation{\omega},$
(143)

and we have the following thermodynamics

$\displaystyle M$
$\displaystyle=\frac{\Sigma_{1}r_{+}^{2}}{L^{2}}\left[\zeta+\frac{\alpha_{2}^{3}}{972\alpha_{3}^{2}}\right],\qquad
S=4\pi\Sigma_{1}r_{+}\Bigg{[}1+\frac{\alpha_{2}^{2}}{18\alpha_{3}}\Bigg{]},$
(144) $\displaystyle 0$
$\displaystyle=S+2\beta\left[-\zeta\mathcal{A}_{0}+\alpha_{2}\mathcal{A}_{2}+2\alpha_{3}\mathcal{A}_{3}\right],\quad\mathcal{A}_{0}=\frac{\Sigma_{1}r_{+}^{2}}{L^{2}},\quad\mathcal{A}_{2}=\mathcal{A}_{3}=0,$
(145) $\displaystyle\beta\variation M$ $\displaystyle=\variation
S+\beta\left(\mathcal{A}_{0}\variation\zeta+\mathcal{A}_{2}\variation\alpha_{2}+\mathcal{A}_{3}\variation\alpha_{3}\right).$
(146)

The Smarr relation in the form (134) follows from the following Noether charge

$Q=\frac{\beta\Sigma_{1}r^{2}}{486\alpha_{3}^{2}L^{2}}\left[972\zeta\alpha_{3}^{2}+\alpha_{2}^{2}-54\alpha_{3}(\alpha_{2}^{2}+18\alpha_{3})f\right].$
(147)

For $\alpha_{3}=0$ in the action (124), the only solution for the scalar field
is $\phi(r)=\text{constant}$ and one obtains the usual thermodynamics of the
BTZ black hole in 3D GR with a negative cosmological constant Hennigar:2020fkv
which can be obtained by setting $\alpha_{2}=0$ in the thermodynamical
relations above.

##  4 Summary

In this paper, we have studied the thermodynamics of the static black hole
solutions satisfying $g_{tt}g_{rr}=-1$ in lower-dimensional Lovelock gravity
up to cubic term. For a comparison with the higher-dimensional case, we
derived the previous results in higher dimensions where the Lovelock gravity
is a pure gravity theory and shed light on a new feature. Without directly
solving for the metric function, it is possible to deduce the following
properties:

  * •

For planar black holes ($k=0$), the thermodynamics takes a universal form that
one has the same relations as in GR Cai:2003kt .

  * •

In $D=4,6$ where the 2nd- and the 3rd-order Lovelock Lagrangians
$\mathcal{L}_{2,3}$ are boundary terms respectively, the metric function does
not get modified due to these terms but the entropy gets a contribution from
the boundary terms (when $k\neq 0$) Cai:2003kt .

  * •

By considering the variation of the couplings, one can establish an extended
first law with potentials conjugate to the couplings and it is possible to
obtain a Smarr relation containing these potentials Kastor:2010gq .

  * •

The reduced action that gives rise to the solutions that we consider
consistently possesses a global scaling symmetry, from which it is possible to
derive the Smarr relation which is compatible with the extended 1st law. To
our knowledge, the fact that this is true for solutions with non-planar
horizons ($k\neq 0$) (beyond GR) has not been noticed before.

After a detailed examination of the lower-dimensional case, we reach the
following conclusions

  * •

The universality of the thermodynamics of planar black holes still holds.

  * •

In $D=4,6$, both the metric function and the entropy get modified.

  * •

The extended 1st law still holds.

  * •

The reduced action in lower dimensions is not invariant under the global
scaling symmetry (when $k\neq 0$). However; following the procedure described
in Ahn:2015shg  to deal with such cases, we have shown that, for the solutions
that we consider, there is a conserved Noether charge giving rise to the Smarr
relation in $D=3,5$. In $D=4,6$, the contribution to the entropy from the
Lagrangians $\mathcal{L}_{2,3}^{\text{ST}}$ come with a $\log(r_{+})$ term,
which breaks the homogeneity. As a result, one obtains a Smarr-like relation
which can still be obtained from the Noether charge but contains modified
potentials that are related to the potentials in the extended 1st law.

Note Added After we submitted our paper to arXiv, it was argued in
Kubiznak:2023xbg  that logarithmic correction to the entropy when
$D=4,\alpha_{3}=0,k\neq 1$ case discussed in Subsection 3.3 originate from
ignoring the shift symmetry of the scalar in the field equations of the
theory, which can be recovered at the action level by adding a certain
boundary term. With this modification of the action, while the entropy simply
obeys the area law, the first law and Smarr relation are satisfied with the
same potentials but with a modified temperature that deviates from the usual
relation to the surface gravity. This procedure might yield similar results
for the $D=6,k\neq 1$ discussed in Subsection 3.1. Additionally, the authors
observe that the entropy obeys the area law in higher-dimensional Lovelock
theories if the temperature is modified similar to the 4D case. In our work,
the Euclidean reduced actions that we have used to study $D=4,6$ cases possess
shift symmetry after integration by parts and we find the above-mentioned
results with the usual relation of the temperature and the surface gravity. We
refer the reader to Kubiznak:2023xbg  for a detailed comparison.

## Acknowledgement

G. D. O. is supported by TÜBİTAK Grant No 118C587. We thank Marek Liška for
useful correspondence on Kubiznak:2023xbg .

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