# On the existence of thermodynamically stable asymptotically flat black holes

Dumitru Astefanesei,(1) Romina Ballesteros,(1,2) Paulina Cabrera,(1)
Gonzálo Casanova(1) and Raúl Rojas(3)

(1)Pontificia Universidad Católica de Valparaíso, Instituto de Física
Av. Brasil 2950, Valparaíso, Chile
(2)Instituto de Física Teórica UAM/CSIC,
C/ Nicolás Cabrera, 13–15, C.U. Cantoblanco, E-28049 Madrid, Spain
(3)Departamento de Física, Universidad de Concepción, Casilla 160-C,
Concepción, Chile

###### Abstract

We use the quasilocal formalism of Brown and York, supplemented with
counterterms, to investigate the thermodynamics of asymptotically flat black
holes. We consider two families of exact regular black hole solutions, which
are thermodynamically stable. The first one consists of four-dimensional
static charged hairy black holes in extended supergravity. The second family
consists of five-dimensional static charged black holes in Gauss-Bonnet (GB)
gravity. Despite the fact that their characteristics are completely different,
we found a striking similarity between their thermodynamic behaviour.

###### Contents

  1. 1 Introduction
  2. 2 Black holes in Einstein-Maxwell-dilaton theory
    1. 2.1 Euclidean action and thermodynamics
    2. 2.2 Equation of state $\Phi=\Phi(Q,T)$
    3. 2.3 Phase diagram and local stability
      1. 2.3.1 Canonical ensemble: $Q$ fixed
      2. 2.3.2 Grand canonical ensemble: $\Phi$ fixed
  3. 3 Black holes in Einstein-Maxwell-Gauss-Bonnet
    1. 3.1 Euclidean action and thermodynamics
    2. 3.2 Equation of state: $Q-\Phi$
    3. 3.3 Phase diagram and local stability
      1. 3.3.1 Canonical ensemble: $Q$ fixed
      2. 3.3.2 Grand canonical ensemble: $\Phi$ fixed
  4. 4 Discussion
  5. 5 Acknowledgments
  6. A Local stability conditions
  7. B Dilaton counterterms in AdS vs flat spacetime

##  1 Introduction

The black hole represents the equilibrium end state of gravitational collapse
and the relationship between thermodynamic entropy and the area of an event
horizon is one of the most robust results in gravitational physics. Once
quantum effects were taken into account, it was understood by Hawking [1] that
black holes can emit radiation. While the Hawking radiation produces a very
small effect that is not relevant from an empirical point of view, the black
hole thermodynamics makes sense only in a theoretical framework in which the
black holes are thermodynamically stable. However, this is not the case for
black holes in flat spacetime, e.g. Schwarzschild, Reissner-Nordström (RN),
and Kerr black holes are thermodynamically unstable.

One way to circumvent this situation is to enclose the asymptotically flat
black hole by a finite ‘box’ [2] such that the radiation can not escape to the
asymptotic region. However, while this proposal is interesting from a
theoretical point of view, it does not provide any concrete hint of physical
situations where this kind of boundary conditions can appear naturally. A more
general construction, for a theory with a non-trivial negative cosmological
constant, was considered by Hawking and Page [3]. They found that, indeed,
there exists a general family of ‘large’ black holes (with the horizon radius
comparable with the radius of anti-de Sitter (AdS) spacetime) that are
thermodynamically stable. In this case, the conformal boundary of AdS (where
the boundary conditions should be imposed) plays the role of the ‘box’.
However, the boundary conditions are changed to correspond to a spacetime with
negative cosmological constant and, while this is a very important result in
the context of AdS-CFT duality [4], it can not be applied to asymptotically
flat black holes.

Therefore, to obtain a well defined thermodynamic framework, the key point is
the existence of asymptotically flat black holes, which are thermodynamically
stable, without imposing artificially boundary conditions corresponding to a
finite box. Elucidating this aspect could be relevant for understanding the
existence of supermassive black holes that can not be formed by gravitational
collapse. One can imagine a scenario where, in the early universe, small black
holes exist in a suitable environment that makes them thermodynamically
stable, e.g. surrounded by dark matter. Then, after a long time, they can grow
up to become supermassive if there is enough matter around they can absorb.

In this work, we carefully investigate the existence of such an ‘environment’
and present a detailed analysis of the thermodynamic stability by using the
quasilocal formalism supplemented with boundary counterterms. Both Gauss-
Bonnet-Maxwell and Einstein-Maxwell-dilaton theories can be interpreted as
particular completions of Einstein-Maxwell gravity, and so the question of how
these theories modify some concrete features of black holes could be of
significant interest. First, we consider a four-dimensional theory, with a
scalar field and its self-interaction, in which there exist exact hairy black
hole solutions. While this is a toy model for static black holes, it shows
that, even in a simple theoretical set up, the hair can stabilize, dynamically
and thermodynamically, the black holes (see, also, [5, 6, 7]). It can very
well be that more complicated models for dark matter can have a similar
effect. Second, we consider the five-dimensional gravity when the Einstein-
Hilbert action is supplemented by GB corrections. These models can be
understood, for example, as effective theories from string theory where the
higher derivative corrections are tree level, depending of the string tension
(there are also quantum corrections depending of the string coupling). It was
found long time ago that there exist exact static neutral black hole solutions
and they are thermodynamically stable [8] (see also [9, 10]).111There is
another physical mechanism that could be important for the stability of black
holes in theories with higher derivative terms, namely the scalarization and
existence of the scalar condensates [11]. We consider the generalization of RN
black hole in GB theory, compute the conserved charges within the quasilocal
formalism of Brown and York [12] that is self-consistent and so we do not need
Wald formalism [13] to obtain the entropy. We find again that there exist
thermodynamically stable charged static black holes.

The quasilocal formalism supplemented with counterterms in flat spacetime was
used in [14] to obtain the thermodynamics of the dipole black ring [15].
However, this formalism was put on a firm ground by Mann and Marolf in [16],
where they have shown that the addition of an appropriate covariant boundary
term to the gravitational action yields a well-defined variational principle
for asymptotically flat spacetimes. This becomes the standard tool that leads
to a natural definition of conserved quantities at spatial infinity, e.g. [17,
18, 19, 20, 21, 22, 23]. In this work, we use concrete counterterms to
circumvent the problem with the background subtraction method when the
‘background’ can not be explicitly constructed. We construct the counterterm
for GB gravity (and fix the ambiguity of an overall numerical factor), which
is consistent with a correct variational principle and regularizes the
Euclidean action. Interestingly, unlike the AdS case, there is no need of
adding counterterms for the scalar field in flat spacetime when it vanishes at
the boundary. We shall consider both the grand canonical ensemble, in which
the electric potential difference between the event horizon and the infinity
is fixed, and the canonical ensemble, in which the electric charge is fixed.

The remainder of the paper is organized as follows: in Section 2, we present
the exact asymptotically flat hairy charged black hole solution and study in
detail the local and global thermodynamic stability. We use the counterterm
method to obtain the on-shell regularized Euclidean action and the quasilocal
formalism to obtain the conserved charges. We identify the region of the phase
space where there exist thermodynamically stable black holes. In Section 3, we
present the charged black hole solution in GB gravity. We obtain the on-shell
Euclidean action, conserved charges, Smarr formula, and verify the quantum-
statistical relation. Again, we identify the region of the phase space where
there exist thermodynamically stable black holes. In the last section, we
present a brief review of our main results, complement the analysis of
thermodynamic behaviour with an analysis using the thermodynamic potential,
and discuss to some extent the extremal limit that is important for the
existence of a consistent canonical ensemble. In Appendix A, we summarizes the
general thermodynamic stability conditions of response functions for charged
black holes. Finally, in Appendix B, we compare the counterterm method in AdS
with the one in flat spacetime and explain why there is no need of scalar
counterterms for regularizing the Euclidean action of asymptotically flat
hairy black holes.

##  2 Black holes in Einstein-Maxwell-dilaton theory

Let us consider the theory given by the action

$I=\frac{1}{2\kappa}\int_{\mathcal{M}}{d^{4}x\sqrt{-g}\left[R-\frac{1}{2}(\partial\phi)^{2}-e^{\phi}{F^{2}}-V(\phi)\right]}$
(1)

where $\kappa=8\pi$, in the unit system where $G=c=1$;
$F^{2}=F_{\mu\nu}F^{\mu\nu}$, where
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the gauge field
and $A_{\mu}$ the gauge potential, and $(\partial\phi)^{2}\equiv
g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$.

The potential [24, 25, 26] is

$V(\phi)\equiv 2\Upsilon\left(2\phi+\phi\cosh\phi-3\sinh\phi\right)$ (2)

where $\Upsilon$ is a dimensionful parameter that characterize the strength of
the potential. Importantly, now it is well understood that in fact this is a
model of a dilaton and its self-interaction, in the sense that the dilaton is
endowed with a potential that originates from an electromagnetic Fayet-
Iliopoulos (FI) term in $\mathcal{N}=2$ extended supergravity in four
spacetime dimensions [27, 28] (see, also, [29]). Therefore, the theory we
consider is consistent and its ground state is stable (recently, exact hairy
charged soliton solutions were constructed in [31, 32, 33, 30]). The schematic
behaviour of the potential (2) is depicted in Fig. 1. The potential is
symmetric under $\phi\rightarrow-\phi$ and $\Upsilon\rightarrow-\Upsilon$
transformations.

Figure 1: Schematic behaviour of the scalar field potential for the family
$\Upsilon<0$ and $\Upsilon>0$. The region $\phi<0$ corresponds to the negative
branch and $\phi>0$ corresponds to the positive branch.

The exact static hairy black hole solution was obtained in [26],

$ds^{2}=\Omega(x)\left[-f(x)dt^{2}+\frac{\eta^{2}dx^{2}}{x^{2}f(x)}+d\Sigma_{2}^{2}\right]\,,\qquad
A_{\mu}=\left[\Phi-\frac{q(x-1)}{x}\right]\delta_{\mu}^{t}\,,\qquad\phi(x)=\ln(x)$
(3)

where $d\Sigma_{2}^{2}\equiv d\theta^{2}+\sin^{2}\theta d\varphi^{2}$ and

$f(x)=\Upsilon\left(\frac{x^{2}-1}{2x}-\ln{x}\right)+\frac{\eta^{2}(x-1)^{2}}{x}\left[1-\frac{2q^{2}(x-1)}{x}\right]\,,\qquad\Omega(x)=\frac{x}{\eta^{2}(x-1)^{2}}$
(4)

In the expressions above, $x$ is a dimensionless coordinate that is related to
the standard canonical radial coordinate, $r$, by the transformation
$\Omega(x)=r^{2}$. There are two constants of integration, $\eta$ and $q$,
which are related to the mass and electric charge of the solution, and $\Phi$
in the expression of gauge field is an arbitrary additive constant. The black
hole event horizon is located at $x=x_{+}$ such that $f(x_{+})=0$ and $x_{+}$
is the biggest root. The asymptotic region is located at $x\rightarrow 1$,
where the conformal factor $\Omega$ diverges, and the scalar field (along with
its potential) vanishes.

We distinguish two families of solutions: the family for which $\Upsilon>0$
and the family for which $\Upsilon<0$. Within each family, there are two
branches of solutions, characterized by the domain for the coordinate $x$,
which in turn fixes the sign of $\phi$. These two branches are characterized
by distinct boundary condition for the scalar field. To obtain the asymptotic
behaviour of the dilaton, we consider the transformation to the canonical
radial coordinate, namely $\Omega(x)=r^{2}$ near the boundary. This gives rise
to two possible changes of coordinates,

$x=1\pm\frac{1}{\eta
r}+\frac{1}{2\eta^{2}r^{2}}\pm\frac{1}{8\eta^{3}r^{3}}+\mathcal{O}(r^{-4})$
(5)

The negative branch corresponds to the case where $0\leq x<1$ (and so
$\phi<0$). By using the corresponding change of coordinates, we obtain the
following boundary expansion: $\phi=-\frac{\eta}{r}+\mathcal{O}(r^{-3})$. The
positive branch corresponds to the case where $1<x\leq\infty$ (and so
$\phi>0$), and the boundary expansion of the scalar field becomes
$\phi=+\frac{\eta}{r}+\mathcal{O}(r^{-3})$.222Generally, for example in string
theory where the scalar fields are moduli related to the coupling constants,
the expansion of the scalar field in flat spacetime is
$\phi=\phi_{\infty}+\Sigma/r+...$, where $\Sigma$ is the scalar charge. A
discussion of why scalar charges (which are not conserved charges) [34], when
$\phi_{\infty}$ can vary, do not appear in the first law of black hole
thermodynamics was presented in [35, 36]. However, since the theory we are
interested in contains the dilaton potential, the boundary conditions for the
dilaton are such that $\phi_{\infty}=0$ (see, also, [37]).

In the remaining of this section, we will be focusing on the family
$\Upsilon>0$, which contains thermodynamically stable black holes. The family
$\Upsilon<0$ only supports black hole solutions for the positive branch, but
in this case the potential (2) becomes unbounded from below, as shown in Fig.
1.

###  2.1 Euclidean action and thermodynamics

Let us briefly present the computation of the on-shell Euclidean action. For
concreteness, we perform the computations in the positive branch, that is,
under the assumption that $x-1>0$.

Both the bulk part of the action and the Gibbons-Hawking boundary term [38]
are

$\displaystyle I^{E}_{bulk}+I_{GH}^{E}=\beta(-TS-\Phi
Q)+\frac{4\pi\beta}{\kappa}\left[\frac{2}{\eta(x-1)}+\frac{\Upsilon-12\eta^{2}q^{2}+3\eta^{2}}{3\eta^{3}}+\mathcal{O}(x-1)\right]$
(6)

where the temperature and entropy are defined as usual: the temperature is
obtained by removing the conical singularity in the Euclidean section, and the
entropy is one quarter of the area of the event horizon,

$T=-\frac{x_{+}}{4\pi\eta}\left.\frac{df(r)}{dr}\right|_{x=x_{+}}\,,\qquad
S=\pi\Omega(x_{+})$ (7)

and $Q$ and $\Phi$ are the electric charge and its conjugate potential. The
electric charge, in terms of the constants of integration, is obtained by
using the Gauss law

$Q=\frac{1}{4\pi}\oint_{s_{\infty}^{2}}{e^{\phi}*F}=\frac{q}{\eta}$ (8)

and the conjugate potential is defined as

$\Phi=A_{t}(x\rightarrow 1)-A_{t}(x_{+})=\frac{q(x_{+}-1)}{x_{+}}$ (9)

Now, in order to remove the divergent contribution appearing in (6), it is
necessary to add a gravitational counterterm for asymptotically flat spacetime
[39, 40, 41, 16, 14]333Unlike AdS spacetime where counterterms for the scalar
field should be also included, in our case there is no need of counterterms
associated to the dilaton. We discuss this point in great detail in Appendix
B.

$I_{ct}=-\frac{1}{\kappa}\int_{\partial\mathcal{M}}{d^{3}x\sqrt{-h}\sqrt{2\mathcal{R}^{(3)}}}$
(10)

where $\mathcal{R}^{(3)}$ is the Ricci scalar on the hypersurface $x=const$.
With this foliation of the spacetime, the boundary $\partial\mathcal{M}$
consists of the hypersurface $x=const$ in the limit $x\rightarrow 1$. In the
Euclidean section, we have

$\displaystyle
I_{ct}^{E}=\frac{4\pi\beta}{\kappa}\left[-\frac{2}{\eta(x-1)}-\frac{\Upsilon-12\eta^{2}q^{2}+6\eta^{2}}{6\eta^{3}}+\mathcal{O}(x-1)\right]$
(11)

The total action is therefore

$I^{E}=I^{E}_{bulk}+I_{GH}^{E}+I_{ct}^{E}=\beta(-TS-\Phi
Q)+\beta\left(\frac{12\eta^{2}q^{2}-\Upsilon}{12\eta^{3}}\right)$ (12)

The last term in (12) is indeed the mass of the black hole, as follows from
computing the conserved charges of the system. We compute this energy by using
the Brown-York quasilocal formalism [12], with the quasilocal boundary stress
tensor

$\tau_{ab}\equiv-\frac{2}{\sqrt{-h}}\frac{\delta I}{\delta h^{ab}}$ (13)

where $I=I_{bulk}+I_{GH}+I_{ct}$, and the index $a$ stands for the coordinates
on the hypersurface $x=const$. For this case, the concrete expression for
$\tau_{ab}$ is [14]

$\tau_{ab}=\frac{1}{\kappa}\left[K_{ab}-h_{ab}K-\Psi\left(\mathcal{R}^{(3)}_{ab}-\mathcal{R}^{(3)}h_{ab}\right)-h_{ab}\Box\Psi+\Psi_{;ab}\right]\,,\qquad\Psi\equiv\left(\frac{2}{\mathcal{R}^{(3)}}\right)^{\frac{1}{2}}$
(14)

Now, according to the Brown-York formalism, the conserved charge associated to
the isometry generated by the killing vector $\xi^{a}$ is

$E=\oint_{s_{\infty}^{2}}{d^{2}\sigma\sqrt{\sigma}N^{a}\xi^{b}\tau_{ab}}$ (15)

where $N^{a}$ is the timelike unit normal to the hypersurface $x=const$ and
$\sigma$ is the determinant of the metric with $x=const$ and $t=const$. The
integration is performed in the limit $x\rightarrow 1$. For
$\xi^{a}=\delta^{a}_{t}$ and the result is

$E=\frac{12\eta^{2}q^{2}-\Upsilon}{12\eta^{3}}$ (16)

that also reproduces the mass read off from the expansion, in the canonical
coordinates, of $g_{tt}$ in the asymptotic region. Therefore, the Euclidean
on-shell action satisfies the quantum-statistical relation

$\frac{I^{E}}{\beta}=E-TS-\Phi Q$ (17)

The computation made so far has assumed that the boundary condition for the
gauge field is $\delta A_{t}|_{\partial\mathcal{M}}=0$. This implies that the
conjugate potential, $\Phi$, is fixed. Thus, the thermodynamic potential
obtained, namely, $\mathcal{G}\equiv E-TS-\Phi Q$, corresponds to the grand
canonical ensemble. One can construct the canonical ensemble, in which $Q$ is
fixed instead, by performing a Legendre transform in ($\Phi$, $Q$), which is
equivalent to adding to the action the following boundary term

$I_{A}=\frac{2}{\kappa}\int_{\partial\mathcal{M}}{d^{3}x\sqrt{-h}\,n_{\nu}F^{\mu\nu}A_{\nu}}$
(18)

giving rise to the thermodynamic potential $\mathcal{F}\equiv E-TS$.

Now that we have all the thermodynamic quantities consistently computed, one
can verify that the first law of black hole thermodynamics is satisfied,

$dE=TdS+\Phi dQ$ (19)

###  2.2 Equation of state $\Phi=\Phi(Q,T)$

To proceed further with the thermodynamic behaviour, let us construct the
equation of state $\Phi=\Phi(Q,T)$. First, we replace $q=Q\eta$ in the
expressions for the thermodynamic quantities, as follows from (8). We then
have the parametric expressions

$T={\frac{\left(x_{+}-1\right)\left[\left({\eta}^{4}{Q}^{2}-\frac{1}{2}{\eta}^{2}-\frac{1}{4}\Upsilon\right){x_{+}^{2}}+\left({\eta}^{4}{Q}^{2}-\frac{1}{2}{\eta}^{2}+\frac{1}{4}\Upsilon\right)x_{+}-2{\eta}^{4}{Q}^{2}\right]}{2\pi\eta
x_{+}^{2}}}\,,\quad\Phi=\frac{Q\eta(x_{+}-1)}{x_{+}}$ (20)

where $\eta$ must be isolated from the horizon equation $f(x_{+})=0$. In Fig.
2, we have depicted the equation of state for the positive and negative
branch, respectively. In both cases we have considered the family
$\Upsilon>0$.

Figure 2: Equation of state $\Phi-Q$, for $T$ fixed. Left hand side: Positive
branch. Right hand side: Negative branch.

The equation of state contains relevant information on the local stability.
Concretely, the response function

$\epsilon_{T}\equiv\left(\frac{\partial Q}{\partial\Phi}\right)_{T}$ (21)

known as isothermal permittivity is a measure of the stability of the
configurations against small fluctuations of the electric charge (for more
details, see [42, 43]). If $\epsilon_{T}>0$, the system is locally stable
against electric fluctuations. In both cases, the positive and negative
branches contain regions where $\epsilon_{T}>0$. For the negative branch (the
plot at the right hand side of Fig. 2), the region for which $\epsilon_{T}>0$
corresponds, as we are going to show in the next section, to the region where
the second relevant response function, the heat capacity at constant electric
charge, $C_{Q}$ (and also $C_{\Phi}$), is negatively defined, indicating a
thermal instability. This is similar with the thermodynamics of RN black hole
and so these configurations are not locally stable.

The interesting case is for the positive branch (the plot on the left hand
side of Fig. 2), where a novel region with $\epsilon_{T}>0$ develops. This is
the region A, as shown in the plot. It looks a very small region in the phase
space, but it is valid for any $Q>0$ (and $T>0$). Next, we show that in this
particular region, the heat capacity is also positive, thus fulfilling the
conditions for local stability. The regions A, B and C are separated by the
curves characterized by $(\partial\Phi/\partial Q)_{T}=0$ (the dotted curve,
that separates A from B) and $(\partial Q/\partial\Phi)_{T}=0$ (the dashed
curve, that separates B from C).

###  2.3 Phase diagram and local stability

The second response function relevant for the local stability is the heat
capacity, both at fixed electric charge, $C_{Q}$, and at fixed conjugate
potential $C_{\Phi}$,

$C_{Q}\equiv T\left(\frac{\partial S}{\partial T}\right)_{Q}\,,\qquad
C_{\Phi}\equiv T\left(\frac{\partial S}{\partial T}\right)_{\Phi}$ (22)

Thermally stable configurations are those with $C_{Q}>0$, for the canonical
ensemble, and $C_{\Phi}>0$, for the grand canonical (see Appendix A for a
brief summary on the criteria for local stability).

####  2.3.1 Canonical ensemble: $Q$ fixed

In Fig. 3, we have depicted the phase diagram $T-S$ of the canonical ensemble
for the positive (left hand side plot) and negative branch (right hand side
plot), respectively. As commented above, the interesting behaviour occurs for
the positive branch, where we observe that $C_{Q}>0$ for both regions, A and
B. However, only in region A we have, in addition, $\epsilon_{T}>0$ and,
hence, only region A contains fully locally stable configurations, for the
positive branch. On the other hand, the negative branch is characterized by
two regions, according to the sign of $C_{Q}$. These two regions are separated
by exactly the same curve that divides the regions $\epsilon_{T}>0$ and
$\epsilon_{T}<0$ in the $\Phi-Q$ phase space (the dashed curve, for which
$(\partial Q/\partial\Phi)_{T}=0$ and $C_{Q}^{-1}=0$) and, thus, the response
functions $C_{Q}$ and $\epsilon_{T}$ can not be simultaneously positive for
the negative branch.

Figure 3: $T-S$ for fixed $Q$. Left hand side: Positive branch. Right hand
side: Negative branch.

####  2.3.2 Grand canonical ensemble: $\Phi$ fixed

A similar analysis can be done in the grand canonical ensemble. As shown in
Fig. 4, since clearly $C_{\Phi}<0$, the black holes of the negative branch are
not stable. However, for the positive branch, we also observe that, in
particular, the region A is characterized by $C_{\Phi}>0$, which also
corresponds to $\epsilon_{T}>0$ and so there exist black hole solutions, which
are thermodynamically stable (region B corresponds to $\epsilon_{T}<0$ and so
the response functions are not simultaneously positively defined). We would
also like to point out that, for the positive branch, the stable black holes
are also globally stable, as they minimize the thermodynamic potential,
$\mathcal{G}$. We are going to discuss this point in more detail in Discussion
section.

Figure 4: $T-S$ for fixed $\Phi$. Left hand side: Positive branch. Right hand
side: Negative branch.

##  3 Black holes in Einstein-Maxwell-Gauss-Bonnet

Let us now consider the Einstein-Maxwell-Gauss-Bonnet action [8]

$I=\frac{1}{2\kappa}\int_{\mathcal{M}}{d^{5}x\sqrt{-g}\left(R+\frac{1}{4}\alpha\mathcal{R}_{GB}-F^{2}\right)}$
(23)

where $\kappa=8\pi$, in the unit system where $G=c=1$,
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, $A_{\mu}$ is the
gauge potential, $\mathcal{R}_{GB}\equiv
R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\sigma\rho}R^{\mu\nu\sigma\rho}$ is the
GB invariant and $\alpha$ is the coupling constant with GB sector. It is
convenient to distinguish two branches, according to the sign of $\alpha$,
since they have different thermodynamic properties, as shown below.

The equations of motion for this system are

$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\frac{1}{4}\alpha H_{\mu\nu}=\kappa
T_{\mu\nu}^{EM}\,,\qquad\nabla_{\mu}F^{\mu\nu}=0$ (24)

where
$T_{\mu\nu}^{EM}\equiv\frac{1}{4\pi}\left(F_{\mu\alpha}F_{\nu}{}^{\alpha}-\frac{1}{4}g_{\mu\nu}F^{2}\right)$
is the energy-momentum tensor for the electromagnetic field, and

$H_{\mu\nu}\equiv
2R_{\mu\alpha\beta\sigma}R_{\nu}{}^{\alpha\beta\sigma}-4R_{\mu\sigma\nu\rho}R^{\sigma\rho}-4R_{\mu\sigma}R_{\nu}^{\sigma}+2RR_{\mu\nu}-\frac{1}{2}\mathcal{R}_{GB}g_{\mu\nu}$
(25)

The spherically symmetric black hole solution was obtained in [44]

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Sigma_{3}^{2}\,,\qquad
A_{\mu}=\left(\Phi-\frac{\sqrt{3}q}{2r^{2}}\right)\delta_{\mu}^{t}$ (26)

where $d\Sigma^{2}_{3}\equiv d\theta^{2}+\sin^{2}\theta
d\phi^{2}+\sin^{2}\theta\sin^{2}\phi d\psi^{2}$ is the line element of the
unit 3-sphere, $\Phi$ is an additive constant in the expression of gauge
potential, and $q$ is the charge parameter. Working with this ansatz, the
differential equation for $f(r)$ becomes

$\left(r^{2}+\alpha-\alpha
f\right)f^{\prime}+2r\left(f-1\right)+\frac{2q^{2}}{r^{3}}=0$ (27)

and there exists two families characterized by $\epsilon=\pm 1$, which can be
expressed in a compact form as

$f(r)=1+\frac{r^{2}}{\alpha}\left[1+\epsilon\left({1+\frac{2\alpha\mu}{r^{4}}-\frac{2\alpha
q^{2}}{r^{6}}}\right)^{\frac{1}{2}}\right]$ (28)

where $\mu$ is the constant of integration related to the mass. However, only
the family $\epsilon=-1$ is consistent with asymptotic conditions of flat
spacetime, namely $f(r\rightarrow\infty)=1+\mathcal{O}(r^{-2})$. The other
family characterized by $\epsilon=1$ is not asymptotically flat, namely
$f(r\rightarrow\infty)=2r^{2}/\alpha+1+\mathcal{O}(r^{-2})$. Since we are
interested in asymptotically flat spacetime, in what follows we are going to
consider the family $\epsilon=-1$.

As in general relativity, in GB theory the black hole configurations can also
have at most two horizons for which $f(r)=0$,

$r_{\pm}=\frac{1}{2}\left[2\mu-\alpha\pm\sqrt{(2\mu-\alpha)^{2}-16q^{2}}\right]^{\frac{1}{2}}$
(29)

with the outer one, $r_{+}$, corresponding to the event horizon. We would like
to emphasize that the solution is regular when $2\mu-\alpha\geq 0$, otherwise
it becomes a naked singularity. Therefore, the extremal black holes exist when
the constraint $(2\mu-\alpha)^{2}=16q^{2}$ is satisfied. In this case, the two
horizons coincide ($r_{+}=r_{-}$) and so the mass parameter can be written as
$\mu=2r_{+}^{2}+(1/2)\alpha$.

In the next section, we shall explicitly compute the regularized quasilocal
stress tensor and energy. As a consistency check, we prove that, once an
ambiguity in the overall factor that multiplies the counterterm is fixed, the
quasilocal mass matches, indeed, the Arnowitt-Deser-Misner (ADM) mass [45, 46,
47, 48].

###  3.1 Euclidean action and thermodynamics

In this section, we use again the counterterm method and quasilocal formalism
of Brown and York [12] to consistently compute the regularized Euclidean on-
shell action, boundary stress tensor, and energy. We verify that the quantum-
statistical relation and first law of black hole thermodynamics are
consistently satisfied. We also obtain Smarr formula in this non-trivial case.

The regularized action of this theory is

$I^{E}=I^{E}_{{bulk}}+I^{E}_{GH}+I_{ct}^{E}$ (30)

where $I^{E}_{bulk}$ is the bulk part of the action given by (23), $I_{GH}$
and $I_{ct}$ are the Gibbons-Hawking boundary term [49] and the gravitational
counterterm, respectively, with their corresponding extension for the GB
sector,

$\displaystyle I^{E}_{GH}$
$\displaystyle=-\frac{1}{\kappa}\int_{\partial\mathcal{M}}{d^{4}x\sqrt{h^{E}}\left(K+\frac{1}{2}\alpha\mathcal{K}_{GB}\right)}$
(31) $\displaystyle I^{E}_{ct}$
$\displaystyle=\frac{1}{\kappa}\int_{\partial\mathcal{M}}d^{4}x\sqrt{h^{E}}\left(\frac{3}{2}\mathcal{R}\right)^{\frac{1}{2}}\left(1+\frac{j}{9}\alpha\mathcal{R}\right)$
(32)

where $h$ is the determinant of the induced metric $h_{ab}$ on the boundary
$\partial\mathcal{M}$, $K_{ab}=\nabla_{a}n_{b}$ is the extrinsic curvature,
with $n_{a}$ being the normal unit vector to $\partial\mathcal{M}$, $K\equiv
h^{ab}K_{ab}$, and

$\mathcal{K}_{GB}\equiv\left[\frac{2}{3}{K_{ac}K_{db}\left(Kh^{cd}-K^{cd}\right)}+\frac{1}{3}K_{ab}\left(K_{cd}K^{cd}-K^{2}\right)\right]h^{ab}-2\left(\mathcal{R}_{ab}-\frac{1}{2}\mathcal{R}h_{ab}\right)K^{ab}$
(33)

and $\mathcal{R}=h^{ab}\mathcal{R}_{ab}$ is the trace of the Ricci tensor on
$\partial\mathcal{M}$.

For now, we use an arbitrary factor denoted by $j$ in the GB counterterm
because the variational principle is well defined for any $j$. This ambiguity
can be eliminated on physical grounds, and we are going to fix this factor in
a consistent manner later on so that the physical quantities are well defined.
A similar couterterm was used in [50] (see, also, [51, 52] for GB counterterms
in AdS).

The on-shell Euclidean action in the bulk can be written as444Here,
$\beta=\int_{0}^{\beta}{d\tau}$ is the periodicity of the imaginary time in
the Euclidean section.

$I^{E}_{\text{bulk}}=\frac{\pi\beta}{8}\left.\left[r^{3}f^{\prime}-{3\alpha}{r}\left(f-1\right)f^{\prime}\right]\right|_{r_{+}}^{r_{b}}-\frac{3\pi\beta
q^{2}}{4r_{+}^{2}}$ (34)

where $f^{\prime}\equiv df/dr$ and $r_{b}$ is the location of the boundary. We
shall take the limit $r_{b}\rightarrow\infty$ at the end of the computation.
Specifically, for RNGB black hole, the result for the bulk term can be written
in the compact form as

$I^{E}_{\text{bulk}}=\frac{\pi\beta}{4}\left(\mu-\frac{r_{+}^{4}+3\alpha
r_{+}^{2}+2q^{2}}{r_{+}^{2}+\alpha}\right)$ (35)

The Gibbons-Hawking boundary term is555Some intermediate results are
$\mathcal{J}_{ab}h^{ab}=-\frac{f^{\frac{1}{2}}\left(3rf^{\prime}+2f\right)}{r^{3}}\,,\;\;\mathcal{G}_{ab}{K}^{ab}=-\frac{3rf^{\prime}+6f}{2r^{3}f^{\frac{1}{2}}}$
where $\mathcal{G}_{ab}\equiv\mathcal{R}_{ab}-\frac{1}{2}\mathcal{R}h_{ab}.$

$I_{GH}^{E}=\frac{\pi\beta}{2}(\mu-\alpha)-\frac{3\pi\beta}{4}r_{b}^{2}+\mathcal{O}\left(r_{b}^{-2}\right)$
(36)

Observe that $I_{GH}^{E}$ contains a divergent part $\propto{}r_{b}^{2}$,
which does not come from the GB sector. As expected, this divergent quantity
is going to be removed with the gravitational counterterm that turns out to be

$I_{ct}^{E}=-\frac{\pi\beta}{2}\left(\frac{3}{4}\mu-j\alpha\right)+\frac{3\pi\beta}{4}r_{b}^{2}+\mathcal{O}\left(r_{b}^{-2}\right)$
(37)

All the divergent contributions are now cancelled out and the final result for
the Euclidean action is

${I^{E}}=\frac{3\pi\beta}{8}\mu+\frac{\pi\beta}{2}(j-1)\alpha-\frac{\pi\beta}{4}\left(\frac{r_{+}^{4}+3\alpha
r_{+}^{2}+2q^{2}}{r_{+}^{2}+\alpha}\right)$ (38)

Notice that the second term in Eq. (38), proportional to $\alpha$, is a finite
contribution purely due to the GB sector. We shall see below that, for
consistency with the asymptotically flat spacetime boundary conditions, we
have to fix $j=1$ such that we recover the usual ADM mass.

We now compute the quasilocal energy for this system at spatial infinity.
According to the Brown-York formalism,

$E_{quasi}=\int_{s_{\infty}^{3}}{d^{3}x\sqrt{\sigma}n^{a}\tau_{ab}\xi^{b}}$
(39)

is the conserved charge associated with the time symmetry of the metric, given
by the Killing vector $\xi^{a}=\delta_{t}^{a}$, where $\sigma$ is the
determinant of the metric on the 3-sphere
$ds_{\sigma}^{2}=r^{2}d\Sigma_{3}^{2}$,
$n_{a}=\delta_{a}^{t}/\sqrt{-g^{tt}}=f^{\frac{1}{2}}\delta_{a}^{t}$ is the
normal unit to the hypersurface $t=$ constant, and $\tau_{ab}$ is the boundary
stress tensor

$\tau_{ab}\equiv\frac{2}{\sqrt{-h}}\frac{\delta I}{\delta
h^{ab}}=\frac{1}{\kappa}\left[K_{ab}-Kh_{ab}+\frac{1}{2}\alpha\left(\mathcal{\tilde{Q}}_{ab}-\frac{1}{2}\mathcal{\tilde{Q}}h_{ab}\right)\right]-\frac{2}{\kappa}\tilde{\Psi}_{ab}$
(40)

where

$\mathcal{\tilde{Q}}_{ab}\equiv
3\mathcal{J}_{ab}+2K\mathcal{R}_{ab}-4\mathcal{R}_{ac}K^{c}_{b}+\mathcal{R}K_{ab}-2K^{cd}\mathcal{R}_{cadb}\,,\qquad\mathcal{\tilde{Q}}=\mathcal{\tilde{Q}}_{ab}h^{ab}$
(41)

and

$\tilde{\Psi}_{ab}\equiv\frac{d\Psi(\mathcal{R})}{d\mathcal{R}}\mathcal{R}_{ab}-\frac{1}{2}\Psi(\mathcal{R})h_{ab}\,,\qquad\Psi(\mathcal{R})=\left(\frac{3}{2}\mathcal{R}\right)^{\frac{1}{2}}\left[1+\frac{1}{9}{j}\alpha\mathcal{R}\right]$
(42)

The components of the boundary stress tensor are

$\tau_{tt}=\frac{f}{8\pi
r_{b}^{3}}\left[3(r_{b}^{2}+\alpha)f^{\frac{1}{2}}-\alpha
f^{\frac{3}{2}}-\left(2j\alpha+3r_{b}^{2}\right)\right]=-\frac{1}{4\pi
r_{b}^{3}}\left[\frac{3\mu}{4}+(j-1)\alpha\right]+\mathcal{O}(r_{b}^{-5})$
(43)
$\tau_{\theta\theta}=-\frac{4r_{b}f^{\frac{1}{2}}\left(f^{\frac{1}{2}}-1\right)+\left(r_{b}^{2}+\alpha-f\alpha\right)f^{\prime}}{16\pi{}f^{\frac{1}{2}}}=-\frac{\mu^{2}-4q^{2}}{32\pi{}r_{b}^{3}}+\mathcal{O}(r_{b}^{-5})$
(44)

where $f=f(r_{b})$ and
$\tau_{\psi\psi}=\sin^{2}\phi\tau_{\phi\phi}=\sin^{2}\theta\sin^{2}\phi\tau_{\theta\theta}$.

We have now all the necessary ingredients to compute the thermodynamic
quantities. By using the formula (39), we get

$E_{quasi}=\frac{3}{8}\pi\mu+\frac{1}{2}\pi(j-1)\alpha$ (45)

This expression for the energy of RNGB black hole is consistent with the usual
statistical formula obtained from the thermodynamic potential in the grand
canonical ensemble:

$E=\left(\frac{\partial
I^{E}}{\partial\beta}\right)_{\Phi}-\frac{\Phi}{\beta}\left(\frac{\partial
I^{E}}{\partial\Phi}\right)_{\beta}=E_{quasi}$ (46)

A similar computation can be done in the canonical ensemble as
$E=(\partial\tilde{I}^{E}/\partial\beta)_{Q}$, where [53]

$\tilde{I}=I+\frac{2}{\kappa}\int_{\partial\mathcal{M}}{d^{4}x\sqrt{-h}F^{\mu\nu}n_{\mu}A_{\nu}}\quad\rightarrow\quad\tilde{I}^{E}=I^{E}+\beta
Q\Phi$ (47)

This action is compatible with the boundary condition for the gauge field that
fixes the electric charge, which is given by the Gauss law,

$Q=\epsilon_{0}\oint{\star F}=\frac{\sqrt{3}\pi}{2}q$ (48)

and $\Phi\equiv A_{t}(\infty)-A_{t}(r_{+})=\frac{\sqrt{3}}{2r_{+}^{2}}q$ is
the conjugate potential. The Hawking temperature is computed as usual by
removing the conical singularity in the Euclidean section,

$T=\beta^{-1}=\frac{1}{4\pi}f^{\prime}(r_{+})=\frac{r_{+}^{4}-q^{2}}{2\pi{}r_{+}^{3}(r_{+}^{2}+\alpha)}$
(49)

Before comparing the ADM and quasilocal masses, let us first proceed further
with computing the entropy directly from the regularized action

$S=-I^{E}+\beta\left(\frac{\partial
I^{E}}{\partial\beta}\right)_{\Phi}=\frac{1}{2}\pi^{2}r_{+}\left(r_{+}^{2}+3\alpha\right)$
(50)

that follows from statistical mechanics, after using the semi-classical
approximation $\ln Z\approx e^{-I^{E}}$, where $Z$ is the partition function.
The entropy receives a contribution from the GB sector and can be also
computed by Wald formalism [13, 54].

Before fixing $j$, let us first verify the quantum-statistical relation. There
is some hope that we can eliminate the ambiguity related to $j$ by using this
relation, but since the $\tau_{tt}$ component of the boundary stress tensor
has a similar contribution, both contributions cancel each other in the
quantum statistical relation. Concretely, we obtain666We have assumed the
boundary condition $\delta A_{\mu}|_{\partial\mathcal{M}}=0$ for the gauge
potential, that fixes $\Phi$ (the grand canonical ensemble). Similarly,
$\mathcal{F}\equiv
E-TS=\frac{3}{8}\pi\mu+\frac{1}{2}\pi(j-1)\alpha-\frac{1}{2}\pi^{2}r_{+}\left(r_{+}^{2}+3\alpha\right)=\beta^{-1}\tilde{I}^{E}$
for the canonical ensemble.

$\mathcal{G}\equiv E-TS-\Phi
Q=\frac{3}{8}\pi\mu+\frac{1}{2}\pi(j-1)\alpha-\frac{1}{4}\pi\left(\frac{r_{+}^{4}+3\alpha{}r_{+}^{2}+2q^{2}}{r_{+}^{2}+\alpha}\right)=\beta^{-1}I^{E}$
(51)

and so it is satisfied for any $j$.

Therefore, the correct way to fix $j$ is to compare the quasilocal mass at
spatial infinity with the ADM mass. Once $j$ is fixed for one specific
solution, the counterterm can be used to regularize the energy of all regular
solutions with the same boundary conditions. We emphasize that the fall off of
GB term is very fast at the boundary and that is why the asymptotic flat
boundary conditions are permitted.

Let us now prove that the quasilocal mass matches the ADM mass only for $j=1$.
We employ the ADM formalism as follows: first, we consider a generic equation
of motion for GB gravity

$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\frac{1}{4}\alpha H_{\mu\nu}=\kappa
T_{\mu\nu}$ (52)

It is convenient to rewrite it as

$R_{\mu\nu}+\frac{1}{4}\alpha\left(H_{\mu\nu}-\frac{1}{3}Hg_{\mu\nu}\right)=\kappa\bar{T}_{\mu\nu}$
(53)

where $H\equiv g^{\mu\nu}H_{\mu\nu}$ and
$\bar{T}_{\mu\nu}=T_{\mu\nu}-\frac{1}{3}Tg_{\mu\nu}$,
$T=g^{\mu\nu}T_{\mu\nu}$. Now, we expand the left hand side for a small
perturbation, $g_{\mu\nu}=\mathring{g}_{\mu\nu}+h_{\mu\nu}$, where
$|h_{\mu\nu}|\ll 1$ is the perturbation around a background spacetime given by
$\mathring{g}_{\mu\nu}$. The left hand side of (53) expands as
$-\frac{1}{2}\Box h_{\mu\nu}+\mathcal{O}(h^{2})$ when the harmonic gauge
condition is considered,

$\mathring{\nabla}_{\mu}\left(h^{\mu\nu}-\frac{1}{2}\mathring{g}^{\mu\nu}h\right)=0$
(54)

where $h=\mathring{g}^{\mu\nu}h_{\mu\nu}$. Since $|h_{\mu\nu}|\ll 1$, we
consider the non-relativistic limit, under which $\Box
h_{\mu\nu}\approx\nabla^{2}h_{\mu\nu}$, $T\approx-T_{00}$. In this limit, we
have the Poisson equation

$\nabla^{2}h_{\mu\nu}=-2\kappa\bar{T}_{\mu\nu}$ (55)

The solution to (55) can be expressed as

$h_{\mu\nu}(x^{i})=\frac{\kappa}{A}\int{\frac{\bar{T}_{\mu\nu}(y^{i})d^{4}y}{|x-y|^{2}}}$
(56)

where $A=\int_{0}^{\pi}{\sin^{2}\theta d\theta}\int_{0}^{\pi}{\sin\phi
d\phi}\int_{0}^{2\pi}{d\psi}=2\pi^{2}$ is the area of the unit 3-sphere. We
obtain

$\nabla_{x}^{2}h_{\mu\nu}(x^{i})=\frac{\kappa}{A}\int{\bar{T}_{\mu\nu}(y^{i})d^{4}y\nabla_{x}^{2}\left(\frac{1}{|x-y|^{2}}\right)}$
(57)

Notice that

$\int_{V}\nabla^{2}\left(\frac{1}{r^{2}}\right)dV=\oint_{S}{\nabla\left(\frac{1}{r^{2}}\right)dS}=-\frac{2}{r^{3}}\oint_{S}{dS}=-2A$
(58)

where $\oint_{S}dS=Ar^{3}$ is the area of the 3-sphere of radius $r$.
Therefore,

$\nabla_{x}^{2}\left(\frac{1}{|x-y|^{2}}\right)=-2A\delta^{4}(x-y)\quad\rightarrow\quad\nabla_{x}^{2}h_{\mu\nu}(x^{i})=-{2\kappa}\int{\bar{T}_{\mu\nu}(y^{i})\delta^{4}(x-y)d^{4}y}=-2\kappa\bar{T}_{\mu\nu}(x^{i})$
(59)

that is consistent with (56).

Now, to get the ADM mass, $M_{ADM}$, one must identify
$M_{ADM}=\int{T_{00}d^{4}x}$. By expanding $h_{00}$ from (56) for the
asymptotic region, $|x|\gg|y|$, we obtain

$\displaystyle h_{00}(x^{i})$
$\displaystyle=\frac{\kappa}{A}\left[\frac{1}{r^{2}}\int{\bar{T}_{00}d^{4}y}+\mathcal{O}(r^{-4})\right]$
(60)
$\displaystyle=\frac{\kappa}{A}\left[\frac{1}{r^{2}}\int{\left(T_{00}-\frac{1}{3}g_{00}T\right)d^{4}y}+\mathcal{O}(r^{-4})\right]$
(61)
$\displaystyle=\frac{\kappa}{A}\left[\frac{2}{3r^{2}}\int{T_{00}d^{4}y}+\mathcal{O}(r^{-4})\right]$
(62) $\displaystyle=\frac{2\kappa M_{ADM}}{3Ar^{2}}+\mathcal{O}(r^{-4})$ (63)

Therefore, the ADM mass can be directly read by obtaining $h_{00}$ from
expanding $g_{00}$,

$M_{ADM}=\frac{3}{8}\pi r^{2}h_{00}$ (64)

where we have replaced $\kappa=8\pi$ and $A=2\pi^{2}$. For the GB metric (28),
we asymptotically expand $g_{tt}$ to read $h_{00}$ around the flat spacetime,

$g_{00}=-1+\frac{\mu}{r^{2}}+\mathcal{O}(r^{-4})\,,\qquad
h_{00}=\frac{\mu}{r^{2}}$ (65)

Therefore, only for $j=1$ we consistently obtain

$M_{ADM}=E_{quasi}=\frac{3}{8}\pi\mu$ (66)

and so this is the correct counterterm that is going to regularize the
Euclidean action for any solution with the same boundary conditions.

We would also like to emphasize that in the limit $r_{b}\rightarrow\infty$,
the trace of the boundary stress tensor vanishes,

$\tau_{ab}h^{ab}=\frac{3\mu}{16\pi r_{b}^{3}}+\mathcal{O}(r_{b}^{-5})$ (67)

and it is covariantly conserved, $\tau^{ab}{}_{;b}=0$, that is compatible with
our solution with no matter or conical defects at spatial infinity [55].

It is also straightforward to verify the first law of black hole
thermodynamics,

$dE=TdS+\Phi dQ$ (68)

The Smarr formula is

$2E=3TS+2\Phi Q+2\mathcal{B}\alpha$ (69)

where the last term is, in principle, consistent with an extension of the
first law, $dE=TdS+\Phi dQ+\mathcal{B}d\alpha$,

$\mathcal{B}\equiv\left(\frac{\partial
E}{\partial\alpha}\right)_{S,Q}=-\frac{3}{8}\pi\left(\frac{1}{2}+\frac{3r_{+}^{2}-2\mu}{r_{+}^{2}+\alpha}\right)$
(70)

when the parameter $\alpha$ can vary, which is not our study case.

###  3.2 Equation of state: $Q-\Phi$

In this section, we obtain the equation of state and study in detail the
regions where the system achieves local stability.

The equation of state $\Phi=\Phi(Q,T)$ can be implicitly written as

$T=\frac{1}{6}\left(\frac{Q\Phi}{\pi}\right)^{\frac{1}{2}}\frac{3-4\Phi^{2}}{\Phi\pi\alpha+Q}$
(71)

In the limit $\alpha=0$, the equation of state reduces to the one of charged
black hole in general relativity. For this case, it is well known that there
are no locally stable configurations, as $C_{Q}$ and $\epsilon_{T}$ can not be
simultaneously positive.

For the family $\alpha>0$, the isotherms $T\neq 0$ in the $Q-\Phi$ plane are
characterized by being ‘closed’, i.e, starting and ending at $Q=\Phi=0$, as
shown in the first plot of Fig. 5. This indicates that there is one curve, say
$\Phi_{0}=\Phi_{0}(Q)$, along which $\epsilon_{T}=0$, and another curve, say
$\Phi_{\infty}=\Phi_{\infty}(Q)$, along which $\epsilon_{T}$ diverges or,
alternatively, $\epsilon_{T}^{-1}=0$. From the expression of the isothermal
permittivity,

$\epsilon_{T}=\frac{Q\left(12\pi\alpha\Phi^{3}+20Q\Phi^{2}+3\pi\alpha\Phi-3Q\right)}{(4\Phi^{2}-3)(Q-\pi\alpha\Phi)\Phi}$
(72)

it follows that the $\Phi_{0}=\Phi_{0}(Q)$ is given implicitly by the cubic
equation

$12\pi\alpha\Phi_{0}^{3}(Q)+20Q\Phi_{0}^{2}(Q)+3\pi\alpha\Phi_{0}(Q)-3Q=0$
(73)

whereas

$\Phi_{\infty}(Q)=\frac{Q}{\pi\alpha}$ (74)

Figure 5: Equation of state $\Phi-Q$, for fixed $T$. Left hand side:
$\alpha=1$. Right hand side: $\alpha=-1$.

Because $\epsilon_{T}$ changes twice its sign along a given isotherm $T\neq
0$, we can distinguish three regions. Regions A and C contain electrically
stable configurations ($\epsilon_{T}>0$), and region B contains electrically
unstable ones. While region B and C mimic the behaviour for charged black
holes in Einstein-Maxwell gravity, the presence of the GB invariant, moduled
by the constant $\alpha$, is responsible for new stable configurations within
region A, which consist of small black holes characterized by
$\Phi>\frac{Q}{\pi\alpha}$, which is equivalent to $r_{+}^{2}<\alpha$.

Moreover, for $\alpha>0$, there exists a maximum allowed temperature that
depends only on $\alpha$. By looking at the first plot in Fig. 5, we can
observe that, as temperature approaches to its maximum, $T_{max}$, the
corresponding isotherm shrinks to eventually disappears. It can be shown, but
also follows from visual inspection, that in the limit $T\rightarrow T_{max}$,
$\Phi\rightarrow Q/(\pi\alpha)$. In other words, the isotherms tend to align
with the dotted curve in Fig. 5. By replacing $\Phi=Q/(\pi\alpha)$ in the
equation of state, then, taking the limit $Q\rightarrow 0$ leads to

$T_{max}=\frac{1}{4\pi\sqrt{\alpha}}$ (75)

On the other hand, for $\alpha<0$, there is one special curve in the $Q-\Phi$
phase space, namely, $\Phi_{0}=\Phi_{0}(Q)$, where $\epsilon_{T}=0$. This
situation is depicted in the second plot of Fig. 5. As shown next, for the
$\alpha<0$ case, the region with $\epsilon_{T}>0$ is thermally unstable,
$C_{Q}<0$.

###  3.3 Phase diagram and local stability

####  3.3.1 Canonical ensemble: $Q$ fixed

Let us focus on the case $\alpha>0$. From the equation of state, we have that,
for a given $Q$ within the region A, the electrically stable configuration is
the one that has the largest value of $\Phi$. We now explicitly show that
configurations within region A are also thermally stable. The heat capacity in
this case is

$C_{Q}=T\left(\frac{\partial S}{\partial T}\right)_{Q}=\frac{3\sqrt{\pi
Q\Phi}(Q+\pi\alpha\Phi)^{2}\left(3-4\Phi^{2}\right)}{2\Phi^{2}(12\pi\alpha\Phi^{3}+20Q\Phi^{2}+3\pi\alpha\Phi-3Q)}$
(76)

It is easy to translate the curves that divide region A, B and C, given by
(73) and (74), into the diagrams $T-S$, as shown in the first plot in Fig. 6.
As commented earlier, region A contains small black holes, $r_{+}^{2}<\alpha$,
that is $S<2\pi^{2}\alpha^{\frac{3}{2}}$. These configurations are thermally
stable as $C_{Q}>0$, and thus, are locally thermodynamically stable.

Figure 6: $T-S$ for fixed $Q$. Left hand side: $\alpha=1$. Right hand side:
$\alpha=-1$.

####  3.3.2 Grand canonical ensemble: $\Phi$ fixed

The thermodynamic stability in the grand canonical ensemble can be
investigated in the same way, though the analysis is simpler. The heat
capacity for this ensemble is

$C_{\Phi}=\frac{3\sqrt{\pi
Q}(Q+\pi\alpha\Phi)^{2}}{2\Phi^{\frac{3}{2}}(\pi\alpha\Phi-Q)}$ (77)

Consider the case of interest, $\alpha>0$. Since region A is characterized by
$\Phi>Q/(\pi\alpha)$, it immediately follows that $C_{\Phi}>0$ within this
region.

Thus, we conclude that region A contains fully thermodynamically stable
configuration both in the canonical and grand canonical ensembles.

##  4 Discussion

In this paper, we provide examples of thermodynamically stable asymptotically
flat black holes. We have shown that effective theories with the dilaton and
its self-interaction, as well as gravity with GB corrections, can allow for
thermodynamic stability regions in the phase diagram of charged static black
holes.

The conserved charges of asymptotically flat black holes are usually computed
by using Hamiltonian methods [45, 46, 47, 48]. However, supplemented with
counterterms, the quasilocal formalism of Brown and York [12] becomes a
powerful framework for computing conserved quantities in general relativity.
The basic idea in [12] is to define a ‘quasilocal’ energy inside a given
finite region that can be directly derived from the gravitational action for
that specific spatially bounded region. The quasilocal energy is the value of
Hamiltonian which generates unit magnitude proper-time translations in a
timelike direction orthogonal to spacelike hypersurfaces at some fixed spatial
boundary and so it agrees with the ADM energy in the limit when the spatial
boundary is pushed to infinity. We emphasize, though, that while within the
ADM formalism the foliation is made by hypersurfaces that are Cauchy surfaces
such that the data on a slice determine completely the future evolution of the
system, that is not necessary the case for the quasilocal formalism. One
important result of our work was to obtain a consistent counterterm for GB
gravity in flat spacetime. While the variational principle is at the basis of
this construction, a relevant subtlety we had to deal with was the existence
of a finite contribution coming from the counterterm that is related to the
ambiguity in defining its overall factor. To fix this ambiguity and construct
the general counterterm that regularizes the action we had to rely on the
Hamiltonian formalism. Once the quasilocal formalism is consistently
supplemented with counterterms, it can be also used for theories with higher
derivative corrections. We emphasize that, unlike in the case of Wald
formalism, we do not need to use the first law a priori and so the quasilocal
formalism is self-consistent providing all information about the thermodynamic
behaviour of the gravitational system.

The existence of asymptotically flat hairy black holes in theories with a
scalar field potential was proposed in [56]. We made this proposal concrete by
analyzing exact solutions when the dilaton together with its potential are
present in the theory. Since in flat spacetime, the effective potential
(obtained from the self-interaction of dilaton together with the non-trvial
coupling to the gauge field) plays the role of the local ‘box’, one can expect
that there can exist thermodynamcally stable black holes in such theories.
This is indeed true for 4-dimensional supergravity theory with FI terms for
which, interestingly, there exist exact hairy solutions. Unlike the black
holes in AdS spacetime, the small black holes (the parameter $\Upsilon$ coming
from the FI sector provide a scale in the theory) are stable. This can be
understood as follows: when the horizon radius is large, the dilaton potential
gets weaker (it vanishes at the boundary) and so the large black holes are not
stable, while for small ones, the self-interaction becomes relevant acting
like a box allowing configurations in stable thermal equilibrium. There exists
a family of solutions that contains two distinct branches, but only one branch
contains thermodynamically stable hairy black holes (see the first plot in
Fig. 2 and the first plot in Fig. 3, where the stable black holes reside
within region A, for the positive branch). The range for which these black
holes exist is close to the extremality.

Surprisingly, the charged black holes in a gravity theory with GB corrections
have a similar thermodynamic behaviour. As discussed in Section 3, there is
again a family of asymptotically flat black holes with two different branches,
but only one branch contains thermodynamically stable black holes. In this
case, it seems that the GB term in the action behaves as an ‘effective
potential’. This can be explicitly checked by comparing the behaviour in the
asymptotic region of $V(\phi)$ vs GB term:

$V(\phi)={\frac{\Upsilon}{30}}{\phi}^{5}+{\frac{\Upsilon}{630}}{\phi}^{7}+\mathcal{O}\left({\phi}^{9}\right)=\frac{\Upsilon}{30\eta^{5}r^{5}}-\frac{3\Upsilon}{560\eta^{7}r^{7}}+\mathcal{O}(r^{-9})$
(78)

and

$\mathcal{R}_{GB}=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\sigma\rho}R^{\mu\nu\sigma\rho}=\frac{72\mu^{2}}{r^{8}}-\frac{360\mu
q^{2}}{r^{10}}-\frac{336\left(\mu^{3}\alpha+q^{4}\right)}{r^{12}}+\mathcal{O}\left(r^{-14}\right)$
(79)

We observe the rapid fall off that explains, in both cases, the existence of
asymptotically flat solutions. However, in both cases, the backreaction deep
in the bulk becomes relevant.

For GB theory, the small black holes in the range close to the extremality are
again thermodynamically stable. However, there is now an extra constraint,
namely the existence of a critical charge: only those with
$Q<\frac{1}{2}\sqrt{3}\pi\alpha$ or, equivalently, $S<2\pi^{2}\alpha^{3/2}$
are thermodynamically stable. This can be also understood from the first plot
in Fig. 5 and Fig. 6, where we see that one zone that contains the near-
extremal black holes (those close to the horizontal dotted curve) is within
Region A (stable), but the other is in Region B (unstable).

To summarize this striking analogy, let us concretely emphasize which branches
are relevant for the existence of thermodynamically stable black hole
configurations. For the Einstein-Maxwell-scalar theory, stability criteria are
met when the scalar field potential has positive concavity and is bounded from
below. This occurs for one of two families, namely the $\Upsilon>0$ one, where
$\Upsilon$ is the (global) parameter that controls the strength of the self-
interaction. Within this family, only the branch for which the scalar field is
positively defined and satisfies some particular boundary condition contains
thermodynamically stable black hole solutions. For charged black holes in the
GB theory, stability criteria are met when the metric is asymptotically flat.
This occurs for one of the two families, namely the one defined by
$\epsilon=-1$, where $\epsilon=\pm 1$ defines the asymptotic structure of the
metric, and, within this family, for one of the two branches, namely the one
where $\alpha>0$, where $\alpha$ controls the strength of the GB correction in
the action.

To complement the thermodynamic analysis in the bulk of the paper that was
based on the equation of state, we now briefly present an analysis of the
thermodynamic potential. We start with the GB case and focus on the case of
interest, namely $\alpha>0$. The relevant information obtained from the
equation of state can also be read off from the $\mathcal{F}-Q$ diagram when
$T$ is fixed, depicted in the first plot of Fig. 7. The locally
thermodynamically stable black holes are the ones in region A, i.e., those
with the biggest value of $\Phi$, for a given
$Q<\frac{1}{2}\sqrt{3}\pi\alpha$. Since
$\Phi=\left(\partial\mathcal{F}/\partial Q\right)_{T}$, the stable
configurations in $\mathcal{F}-Q$ diagram correspond to the lowest part of
each isotherm, having the biggest slope. Also from Fig. 7, it follows that
these configurations minimize the thermodynamic potential. From this
observation, we conclude that locally stable black holes are also globally
stable.

Figure 7: Left hand side: $\mathcal{F}-Q$ for fixed $T$ for $\alpha=1$. Right
hand side: $\mathcal{F}-T$ for fixed $Q$ for $\alpha=1$.

We shall investigate the global stability using the $\mathcal{F}-T$ diagram,
depicted in the second plot of Fig. 7. Since
$C_{Q}=-T(\partial^{2}\mathcal{F}/\partial{T^{2}})_{Q}$, it follows that
locally stable configurations are those satisfying
$(\partial^{2}\mathcal{F}/\partial{T^{2}})_{Q}<0$, which also corresponds to
the lower part of the curve, containing again configurations that minimize
$\mathcal{F}$.

The thermodynamic potential for the hairy black hole solution is depicted in
Fig. 8 where we consider the positive branch. However, the thermodynamically
stable configurations are present only in the positive branch, corresponding
to the first plot in Fig. 8. Since
$C_{Q}=-T(\partial^{2}\mathcal{F}/\partial{T^{2}})_{Q}$, we observe that the
locally stable configurations $C_{Q}>0$ are those with negative concavity,
which correspond to the configurations that minimizes $\mathcal{F}$.

Figure 8: $\mathcal{F}-T$ for fixed $Q$, in the positive branch of the hairy
solutions.

Therefore, in a similar manner as we did in the GB case, we obtain that these
solutions are also globally stable.

Since there is no solution of flat spacetime with constant charge, and unlike
AdS spacetime where hairy charged solitons were explicitly constructed in some
specific cases [31, 32, 33], making sense of the ground state in this case is
not obvious. However, as in [42], one can consider the extremal black hole
(otherwise the spacetime can collapse in a naked singularity) as the state
with respect to which we compute the energy. Therefore, to complete the
analysis, let us consider the extremal limit that is relevant for the
existence of the canonical ensemble. One can use the entropy function
formalism [59, 57, 58] to obtain the near horizon geometry of black hole
solutions in theories that are diffeomorphism and gauge invariant. This method
is based on the existence of an enhanced $AdS_{2}$ in the near horizon
geometry that also plays an important role for understanding the entropy of
spinning [59] astrophysical black holes [60]. For asymptotically flat hairy
black holes in a theory with a dilaton potential, this analysis was done in
[6] (see, also, [61] for a related discussion in a different context) and for
GB charged hairy black holes, the details can be found in [51] and so we do
not repeat them here. We would only like to emphasize that, for the extremal
black holes we have considered in our work, the entropy does not vanish and so
these are regular solutions of the equations of motion. In the GB case, the
results can be obtained analytically, as follows: consider $T=0$ limit, namely
$r_{+}^{2}=q$, as follows from (49). By replacing this value in the horizon
equation, $f(r_{+})=0$ or, equivalently, in (29), we get the following
relation between the conserved charges for the extremal black hole:

$E=\frac{1}{16}\left({3\pi\alpha}+8\sqrt{3}Q\right)$ (80)

and the entropy becomes

$S=\frac{(2\pi Q)^{\frac{1}{2}}}{2\cdot
3^{\frac{1}{4}}}\left(3\pi\alpha+\frac{2Q}{\sqrt{3}}\right)$ (81)

In the hairy case, we can show, through numerical computations, that the
entropy of extremal black holes is also positive, with its values depending of
the electric charge. It turns out that, for increasing values of $Q$, the
entropy of extremal hairy black hole rapidly approaches to zero, as depicted
in Fig. 9. This can be also seen from the first plot of Fig. 3, though, from
that plot and for curves with large values of $Q$, it is not as easy to
discern that $S(T=0)>0$.

Figure 9: Entropy of extremal black hole for the hairy black holes in the
positive branch. $\Upsilon=1$.

While the theories are completely different, we would like to point out that,
generally, theories with higher derivatives $f(R)$ are equivalent with
theories with a scalar field and its self-interaction [62, 63]. However, this
consideration can not be applied for our work because one theory is
$4$-dimensional, while the other is $5$-dimensional. The GB term is trivial in
$4$-dimensions, but once it is coupled with the scalar field it contributes to
the equations of motion and so we expect that the hairy black holes receive
corrections. Since we were not able to generate exact solutions, we leave the
detailed analysis of this specific case for a future work.

##  5 Acknowledgments

The work of DA, PC, and GC was supported by the Fondecyt Regular Grant
1242043. The work of RB has been supported by the National Agency for Research
and Development [ANID] Chile, Doctorado Nacional under grant 2021-21211461. RR
thanks the financial support from ANID, by means of FONDECYT Postdoctorado
grant number 3220663.

##  Appendix A Local stability conditions

The local stability of equilibrium configurations follows from demanding that
heat capacity $C_{Q}\equiv T(\partial S/\partial T)_{Q}$ and isothermal
permittivity $\epsilon_{T}\equiv(\partial{Q}/\partial\Phi)_{T}$ to be
simultaneously positive defined. The proof follows from considering that at
stable equilibrium, the entropy has a maximum value with respect to the
entropy of the system when considering the fluctuations. We can use the
argument (see, for instance, [64]) that, if $S(E,Q)$ represents the entropy
before some fluctuations in $E$ and $Q$, given by $\delta E$ and $\delta Q$,
respectively, then, for the configuration $S(E,Q)$ to be stable, the average
entropy $\frac{1}{2}\left[S(E+\delta E,Q+\delta Q)+S(E-\delta E,Q-\delta
Q)\right]$ after the perturbation cannot be greater than the initial one. The
same argument is valid in terms of the energy: for a given state $E=E(S,Q)$ to
be locally stable, the average energy after the perturbation can not be lower
than the initial one. This gives rise to three mathematical conditions for
local stability, namely,

$\left(\frac{\partial^{2}E}{\partial
S^{2}}\right)_{Q}\left(\frac{\partial^{2}E}{\partial
Q^{2}}\right)_{S}-\left[\left(\frac{\partial}{\partial
S}\right)_{Q}\left(\frac{\partial E}{\partial
Q}\right)_{S}\right]^{2}>0\,,\qquad\left(\frac{\partial^{2}E}{\partial
S^{2}}\right)_{Q}>0\,,\qquad\left(\frac{\partial^{2}E}{\partial
Q^{2}}\right)_{S}>0$ (82)

Consider the canonical ensemble, for which the thermodynamic potential is
$\mathcal{F}=E-TS$. For infinitesimal changes in its thermodynamics, we can
describe the system by $d\mathcal{F}(T,Q)=-SdT+\Phi dQ$, and it can be shown,
using the standard thermodynamic relations, that the conditions (82) can be
written in terms of $\mathcal{F}$ and, indeed, can be reduced to only two
independent requirements:

$\displaystyle C_{Q}$ $\displaystyle=T\left(\frac{\partial S}{\partial
T}\right)_{Q}=-T\left(\frac{\partial^{2}\mathcal{F}}{\partial{T}^{2}}\right)_{Q}>0$
(83) $\displaystyle\epsilon_{T}$ $\displaystyle=\left(\frac{\partial
Q}{\partial\Phi}\right)_{T}=\left(\frac{\partial^{2}\mathcal{F}}{\partial{Q^{2}}}\right)_{T}^{-1}>0$
(84)

where we have used $S=-\left({\partial\mathcal{F}}/{\partial{T}}\right)_{Q}$
and $\Phi=\left({\partial\mathcal{F}}/{\partial{Q}}\right)_{T}$.

In the grand canonical ensemble, it can be shown that stability also follows
from analyzing the concavity of the thermodynamic potential $\mathcal{G}$. In
this case, $\mathcal{G}=E-TS-\Phi Q$, the first law takes the form
$d\mathcal{G}(T,\Phi)=-SdT-Qd\Phi$, and the relevant response function for
local stability in this case is

$C_{\Phi}\equiv T\left(\frac{\partial S}{\partial
T}\right)_{\Phi}=-T\left(\frac{\partial^{2}\mathcal{G}}{\partial{T}^{2}}\right)_{\Phi}>0$
(85)

where $S=-\left({\partial\mathcal{G}}/{\partial{T}}\right)_{\Phi}$.

##  Appendix B Dilaton counterterms in AdS vs flat spacetime

Let us consider the Einstein’s equation derived from Einstein-Maxwell-scalar
gravity (1), but now with the general potential (see, e.g., [69] where the
extremal limit was also discussed)

$V(\phi)=\frac{2\Lambda}{3}\left(2+\cosh\phi\right)+2\Upsilon\left(2\phi+\phi\cosh\phi-3\sinh\phi\right)$
(86)

that contains the comological constant $\Lambda$. We now consider the
asymptotically AdS solution, for which the limit $\Lambda\rightarrow 0$ leads
to the asymptotically flat solution,

$ds^{2}=\Omega(x)\left[-f(x)dt^{2}+\frac{\eta^{2}dx^{2}}{x^{2}f(x)}+d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right]\,,\quad
A_{\mu}=\left[\Phi-\frac{q(x-1)}{x}\right]\delta_{\mu}^{t}$ (87)

where

$\Omega(x)=\frac{x}{\eta^{2}(x-1)^{2}}\,,\qquad
f(x)=-\frac{\Lambda}{3}+\Upsilon\,\left({\frac{{x}^{2}-1}{2x}}-\ln{x}\right)+{\frac{{\eta}^{2}\left(x-1\right)^{2}}{x}\left[1-{\frac{2\left(x-1\right){q}^{2}}{x}}\right]}$
(88)

By taking the trace of the Einstein’s equation, we can use the relation
$R=\frac{1}{2}(\partial\phi)^{2}+2V$ to simplify the bulk part of the action
that yields

$I_{bulk}=\frac{1}{2\kappa}\int_{\mathcal{M}}{d^{4}x\sqrt{-g}\left[R-\frac{1}{2}(\partial\phi)^{2}-e^{\phi}{F^{2}}-V(\phi)\right]}=\frac{1}{2\kappa}\int_{\mathcal{M}}{d^{4}x\sqrt{-g}\left[V(\phi)-e^{\phi}{F^{2}}\right]}$
(89)

The on-shell Euclidean action is

$I_{bulk}^{E}=\frac{1}{4}\beta\int_{x_{+}}^{x_{b}}dx\left[\frac{2\eta\Omega(x)}{x}-\frac{(xf(x)\Omega^{\prime}(x))^{\prime}}{\eta}\right]$
(90)

Together, the bulk part and Gibbons-Hawking boundary term add up to

$I_{bulk}^{E}+I_{GH}^{E}=\frac{\beta\left[(12\eta^{2}q^{2}-\Upsilon)(x_{+}-1)-6\eta^{2}(x_{+}-3)\right]}{24\eta^{3}(x_{+}-1)}-\frac{\beta\left(6\eta^{2}-\Lambda\right)}{6\eta^{3}(x_{b}-1)}+\frac{\beta\Lambda}{2\eta^{3}(x_{b}-1)^{2}}+\frac{\beta\Lambda}{3\eta^{3}(x_{b}-1)^{3}}$
(91)

where $x_{b}\rightarrow 1$ is the boundary location. We note that there are
three divergent terms proportional with $\Lambda$ that are not going to
survive for the asymptotically flat solution when the cosmological constant
vanishes. The gravitational counterterm required to remove the divergences is
specific to the flat and AdS spacetimes. For asymptotically flat spacetime,
the gravitational counterterm is

$I_{ct(flat)}^{E}=\frac{1}{\kappa}\int_{\partial\mathcal{M}}{d^{3}x\sqrt{|h|}\sqrt{2\mathcal{R}^{(3)}}}=\frac{\beta}{\eta(x_{b}-1)}-\frac{\beta\left(12\eta^{2}q^{2}-6\eta^{2}-\Upsilon\right)}{12\eta^{3}}+\mathcal{O}(x_{b}-1)$
(92)

It is clear that the counterterm for asymptotically flat spacetime perfectly
cancels the only divergence coming from $I_{bulk}+I_{GH}$ when $\Lambda=0$.
The total action for flat spacetime satisfies the quantum-statistical relation
$\beta^{-1}I^{E}=E-TS-\Phi Q$, as shown in Section 2.

Now, the gravitational counterterm for asymptotically AdS spacetime is

$\displaystyle I_{ct,(AdS)}^{E}$
$\displaystyle=\frac{1}{\kappa}\int_{\partial\mathcal{M}}{d^{3}x\sqrt{h}\left[\frac{2}{\ell}+\frac{\ell}{2}\mathcal{R}^{(3)}\right]}$
(93)
$\displaystyle=\frac{\beta\left(\Lambda+4\Upsilon+24\eta^{2}-48\eta^{2}q^{2}\right)}{48\eta^{3}}+\frac{\beta(8\eta^{2}-\Lambda)}{8\eta^{3}(x_{b}-1)}-\frac{\beta\Lambda}{2\eta^{3}(x_{b}-1)^{2}}-\frac{\beta\Lambda}{3\eta^{3}(x_{b}-1)^{3}}$
(94)

where $\Lambda=-3/\ell^{2}$. For the AdS case, the divergence coming from
$I_{bulk}+I_{GH}$ are not completely cancelled by (94), and one of the terms
$\propto(x_{b}-1)^{-1}$ survives. Concretely,

$I_{bulk}^{E}+I_{GH}^{E}+I_{ct,(AdS)}^{E}=\frac{\beta\left[12{\eta}^{2}(x_{+}+1)+(\Lambda+2\Upsilon-24{\eta}^{2}{q}^{2})(x_{+}-1)\right]}{48\eta^{3}(x_{+}-1)}+\frac{\beta\Lambda}{24\eta^{3}(x_{b}-1)}$
(95)

and the remaining divergent term is cancelled by the contribution from the
scalar field counterterm [65, 66] (consistent with the Hamiltonian formalism
[67, 68])

$I_{\phi}^{E}=\frac{1}{2\kappa}\int_{\partial\mathcal{M}}{d^{3}x\sqrt{h}\left[\frac{\phi^{2}}{2\ell}+\frac{W(\phi)}{\ell
A}\phi^{3}\right]}=-\frac{\beta\Lambda}{24\eta^{3}(x_{b}-1)}-\frac{\beta\Lambda}{48\eta^{3}}$
(96)

where $\phi(r)=A/r+B/r^{2}+\cdots$, $r=\sqrt{\Omega(x)}$, and $B=dW(A)/dB$ (in
this case, $B=0$ and $W=0$). The total action also satisfies the quantum-
statistical relation, $\beta^{-1}I^{E}=E-TS-\Phi Q$, where
$I^{E}=I^{E}_{bulk}+I_{GH}^{E}+I_{ct,(AdS)}^{E}+I^{E}_{\phi}$.

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