# Equilibrium Temperature for Black Holes with Nonextensive Entropy

Ilim Çimdiker<EMAIL_ADDRESS>Institute of Physics, University
of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland  Mariusz P.
<EMAIL_ADDRESS>Institute of Physics, University of
Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland  National Centre for
Nuclear Research, Andrzeja Sołtana 7, 05-400 Otwock, Poland  Copernicus Center
for Interdisciplinary Studies, Szczepańska 1/5, 31-011 Kraków, Poland  Hussain
<EMAIL_ADDRESS>Institute of Physics, University of Szczecin,
Wielkopolska 15, 70-451 Szczecin, Poland

###### Abstract

Hawking temperature has been widely utilised in the literature as the
temperature that corresponds to various nonextensive entropies. In this study,
we analyze the compatibility of the Hawking temperature with the nonextensive
entropies. We demonstrate that, for every nonextensive entropy, one may define
an effective temperature (which we call equilibrium temperature) by utilizing
the equilibrium condition, and that there is always an additive equilibrium
entropy associated with this effective temperature. Except for Bekenstein
entropy, we show that Hawking temperature is thermodynamically inconsistent
with other nonextensive entropies. We focus on the equilibrium requirement for
the Tsallis-Cirto black hole entropy and demonstrate that the Bekenstein-
Hawking entropy is the related equilibrium entropy, and the Hawking
temperature is the associated equilibrium temperature for the Tsallis-Cirto
black hole entropy.

##  1 Introduction

The seminal works of Bekenstein [1], and Hawking [2, 3] on the thermodynamics
of black holes [4, 5, 6, 7] have a wide range of applications in gravitation
and cosmology. These concepts have been applied, for instance, to investigate
gravity from a thermodynamic perspective [8], to derive Einstein’s field
equations from the first law of thermodynamics [9], to study holographic dark
energy [10], and to examine the universe’s accelerated expansion from a
thermodynamic perspective [11, 12, 13, 14]. Numerous studies have been made to
extend these concepts from a thermodynamic and quantum perspective, including
quantum gravity corrections [15, 16, 17, 18] and thermal fluctuation
corrections [19]. Quantum field theory is used to study Hawking radiation by
incorporating quantum effects on the horizon [2, 3, 20, 21]. This enables the
calculation of the Hawking temperature, which supports Bekenstein’s idea of a
black hole’s entropy. This concept of entropy is somewhat geometric and relies
on Hawking’s area theorem [22]. The laws of black hole thermodynamics [7] are
analogous to the laws of classical thermodynamics by defining the Bekenstein-
Hawking entropy $S_{bh}$ [1] and the Hawking temperature $T_{bh}$ [2] as 444
We utilize natural units by taking the speed of light $c$, the Newton’s
constant $G$, the reduced Planck’s constant $\hslash$, and the Boltzmann’s
constant $k_{B}$ equal to one. We have introduced $k_{B}$ in some sections to
check the dimensional consistency.

$S_{bh}=\frac{A}{4},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ T_{bh}=\frac{\kappa}{2\pi}.$ (1)

For the case of a Schwarzschild black hole with mass $M$, the area $A$ becomes
$A=4\pi r_{h}^{2}=16\pi M^{2}$, where $r_{h}=2M$ is the Schwarzschild radius,
and the surface gravity $\kappa$ becomes $\kappa=1/4M$. By using the
quantities in equation (1), the first law of black hole thermodynamics for
Schwarzschild black hole can be written as $dM=\kappa dA/8\pi,$ which is
equivalent to the first law of thermodynamics $dE=TdS-PdV$, except for the
pressure-volume term $PdV$ [23], with mass $M$ playing the role of internal
energy $E$, $\kappa/2\pi$ playing the role of temperature and $A/4$ playing
the role of entropy, respectively. The $PdV$ term can be introduced for an
anti-de Sitter (AdS) black hole by considering the negative cosmological
constant $\Lambda<0$ as pressure $P$ [24, 25, 26, 27], and introducing the
volume $V$ as

$P=-\frac{\Lambda}{8\pi},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ V=\frac{4}{3}\pi r_{h}^{3},$ (2)

so that the extended first law of thermodynamics reads as

$dM=T_{bh}dS_{bh}-PdV.$ (3)

In this way, the Smarr formula [28] for the mass of the black hole reads

$M=2T_{bh}S_{bh}-2PV.$ (4)

Taking the cosmological constant as a thermodynamic pressure provides the
notion of volume for black holes, which is missing from the first law of black
hole thermodynamics. In this scenario, the mass $M$ no longer represents the
black hole’s internal energy. However, it now acts similarly to the
gravitational equivalent of enthalpy, which is the sum of internal energy $E$
and the work term $PV$. Furthermore, there are interesting consequences in
this scenario [26, 29, 30, 31]. For example, black holes act like Van der
Waals fluids. In this manner, intriguing phase behavior such as the reentrant
phase transition and triple points in the context of black holes have been
examined. For more details, see [32] and references therein.

The primary difficulty with the idea of black hole entropy is the absence of a
suitable statistical mechanical explanation. Instead, it must rely on
Bekenstein’s definition, which states that because black hole entropy is
directly proportional to the area of its event horizon rather than its volume,
it is nonadditive and presumed to be nonextensive. Because of this, black
holes and other cosmological and gravitational applications cannot be well
described by classical thermodynamics or statistical mechanics. Instead, many
nonextensive statistical mechanics approaches have been used to examine black
holes and other cosmological applications [33, 34, 35, 36, 37, 38, 39, 40, 41,
42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55]. These nonextensive
approaches include a variety of proposals. Tsallis statistics [56, 57], Rényi
statistics [58], the Tsallis-Cirto [59, 60] and Barrow entropy [61] for the
case of black holes, Sharma-Mittal statistics [62, 63, 64], and Kaniadakis
statistics [65] are a few examples. Long-range forces are taken into account
in Tsallis’ statistical mechanics or thermodynamics, and nonextensivity
results from these long-range interactions, which are ignored in traditional
Gibbs statistics. Rényi statistics are widely used in quantum information, and
Rényi entropy measures the entanglement of quantum systems. Sharma-Mittal
statistics is only an extension of Tsallis and Rényi statistics, whereas
Kaniadakis statistics is inspired by the Lorentz transformation of special
relativity. Barrow entropy is mathematically equal to Tsallis-Cirto entropy in
the case of black holes, but the motivation for it derives from the fractal
structure of the black hole horizon caused by quantum fluctuations. Tsallis-
Cirto black hole entropy is driven to make the Bekenstein entropy additive and
extensive.

For the classical thermodynamic systems in thermal equilibrium, the zeroth law
utilizes the transitivity relationship between systems to define an empirical
temperature for each system. Another method of defining temperature is to use
the equilibrium condition of the system, which maximizes its overall entropy.
In addition to this, entropy and internal energy must adhere to the additive
composition rule. The additivity of entropy corresponds to ignoring long-range
forces, whereas the additivity of internal energy refers to ignoring the
interaction energy between composite systems. The temperatures derived from
the transitivity relation and the equilibrium condition in traditional Gibbs
thermodynamics are equal in this regard. Interestingly, considering a system’s
strong interaction with a thermal bath, such as for a strongly coupled system,
the zeroth law’s criterion is based on equilibrium conditions rather than
transitivity relations [66]. Therefore, there is a clear distinction between
the terms “in equilibrium” and “in thermal equilibrium”. It also implies that
a system may approach equilibrium but not necessarily thermalize when it is
strongly coupled to a bath. This means that the additive composition rule for
both internal energy and entropy is sufficient for thermal equilibrium in
Gibbs thermodynamics.

In nonextensive thermodynamics, the zeroth law has numerous issues, as noted
in [67, 68, 69]. Because long-range forces are considered, entropy does not
obey the additive rule. We can assume that the internal energy is additive by
considering weakly interacting systems. The equilibrium condition provides an
effective equilibrium temperature [70], from which one can calculate the
equilibrium entropy for a system. This equilibrium temperature can be called a
zeroth law compatible temperature in the nonextensive setup because the
absolute temperatures defined for each subsystem differ from the equilibrium
temperatures derived from the equilibrium condition. Like, as for the strongly
coupled quantum systems, the nonextensive systems can be “in equilibrium” at
the effective temperature but not “in thermal equilibrium”. In fact, the
zeroth law in a nonextensive setup can be only defined by using the
equilibrium condition. Similarly, the corresponding equilibrium entropy
differs from the nonextensive one and follows the additive composition rule.

Nonextensive thermodynamics has several applications in cosmology and
gravitation. Most cosmological investigations use nonextensive entropies along
with the Hawking temperature. The consistent thermodynamic quantities that
follow the thermodynamic relations are essential to these applications from a
thermodynamic point of view. For example, Rényi temperature and Rényi entropy
are consistent thermodynamically when employed together, as in the case of
Hawking temperature and Bekenstein entropy. The question now is whether
nonextensive entropies defined on black holes and cosmological horizons are
thermodynamically consistent with Hawking temperature. To answer this
question, this article focuses on a few composition rules for different
definitions of the nonextensive entropy of black holes. We will focus on the
equilibrium conditions and examine the equilibrium temperatures in the
nonextensive setup. We shall also comment on the thermodynamic consistency of
Hawking temperature with nonextensive entropies by using standard
thermodynamic relations. Additionally, we will utilize the Schwarzschild black
hole as an example of a thermodynamic system. In this context, we will not
explore any cosmological models, but the justification and analysis in this
paper will hold for cosmological models as well.

The structure of the paper is as follows. In section 2, we introduce the
Tsallis nonextensive entropy and find the corresponding equilibrium
temperature. In section 3, we apply it to the Schwarzschild black hole and
define its energy, temperature, and mass. In section 4, we investigate the
Tsallis-Cirto black hole entropy and equilibrium conditions and we briefly
mention the relation between the Tsallis-Cirto and Barrow entropies. Finally,
in section 5, we summarize our main conclusions.

##  2 Composition Rule and Equilibrium Temperature in Tsallis Nonextensive
Setup

Tsallis nonextensive entropy generalizes the Gibbs-Shannon’s entropy into [57]

$S_{q}=-k_{B}\sum_{i}[p(i)]^{q}\ln_{q}p(i),$ (5)

where $p(i)$ is the probability distribution defined on a set of microstates
$\Omega$, the parameter $q$ determines the degree of nonextensivity and we
consider it positive to ensure the concavity of $S_{q}$. The q-logarithmic
function $\ln_{q}p$ is defined as

$\ln_{q}p=\frac{p^{1-q}-1}{1-q},$ (6)

such that, in the limit, $q\to 1$, the equation (5) reduces to Gibbs-Shannon’s
entropy

$S_{G}=-k_{B}\sum_{i}p(i)\ln p(i).$ (7)

Note that Tsallis entropy (5) satisfies a nonadditive composition rule, which
we shall discuss in the next section, while Gibbs entropy (7) satisfies the
additive composition rule. However, via “formal logarithm” approach [69], one
can write a corresponding additive entropy in terms of $S_{q}$ such that

$S_{R}=\frac{k_{B}}{1-q}\left[\ln{\left(1+\frac{1-q}{k_{B}}S_{q}\right)}\right],$
(8)

which happens to be the Rényi entropy [58]

$S_{R}=k_{B}\frac{\ln\sum_{i}p^{q}(i)}{1-q}.$ (9)

Later, we shall see that $S_{R}$ is related to the equilibrium condition and
it will be equilibrium entropy for a nonextensive system, which will also
correspond to an equilibrium temperature defined from the equilibrium
condition by maximizing the nonextensive entropy (5).

By following [71], we consider a thermodynamic system composed of two
independent subsystems, $1$ and $2$, in contact with each other. By defining a
general composition rule

$S_{12}=f(S_{1},S_{2}),$ (10)

which tells us that any total entropy $S_{12}$ can be expressed in terms of
the entropies of subsystems $S_{1}$ and $S_{2}$. Here, $f$ is a bivariate
function of the $C^{2}$, and it is assumed to be symmetric. In this context,
the Gibbs-Shannon entropy $S_{G}$ satisfies

$f(S_{G1},S_{G2})=S_{G12}=S_{G1}+S_{G2},$ (11)

and Tsallis nonextensive entropy $S_{q}$ follows the following general
nonadditive composition rule [71]

$S_{q12}=S_{q1}+S_{q2}+\frac{\lambda}{k_{B}}S_{q1}S_{q2},$ (12)

for a thermodynamic system having total entropy $S_{q12}$, which is composed
of two independent subsystems having entropies $S_{q1}$ and $S_{q2}$, which
are in contact with each other thermally. Here, we introduced $\lambda=1-q$.
in above equation (12).

Since we are interested in equilibrium thermodynamics, we consider fixed total
internal energy $U_{q12}=U_{q1}+U_{q2}$ for a composite system, where $U_{q1}$
and $U_{q2}$ are the internal energies of the indivisible subsystems, and the
equilibrium condition can be found by maximizing the equation (12), i.e.,
$\delta S_{q12}=0$ with $\delta U_{q12}=0$, which gives the following
condition

$\frac{k_{B}\frac{\partial S_{q1}}{\partial
U_{q1}}}{1+(\lambda/k_{B})S_{q1}}=\frac{k_{B}\frac{\partial S_{q2}}{\partial
U_{q2}}}{1+(\lambda/k_{B})S_{q2}}=k_{B}\beta^{*},$ (13)

where $k_{B}\beta^{*}$ is a separation constant and we introduced $\beta$ for
each subsystem, which is defined as

$k_{B}\beta=\frac{\partial S_{q}}{\partial U_{q}}.$ (14)

Now we can easily write down the effective temperature as an equilibrium
temperature by using the equilibrium condition (13) such that

$T_{eq}=\frac{1}{k_{B}\beta^{*}}=(1+\frac{\lambda}{k_{B}}S_{q})\frac{1}{k_{B}\beta}.$
(15)

Similarly, the equilibrium pressure $P_{eq}$ can be defined in the state of
mechanical equilibrium by maximizing the entropy (12) with fixed total volume
$V=V_{1}+V_{2}$ of the composite system and individual subsystem volumes
$V_{1}$ and $V_{2}$, which gives the following condition

$\frac{\partial S_{q1}/\partial V_{1}}{1+(\lambda/k_{B})S_{q1}}=\frac{\partial
S_{q2}/\partial V_{2}}{1+(\lambda/k_{B})S_{q2}}=\frac{P_{eq}}{T_{eq}},$ (16)

so that the physical pressure reads as

$P_{eq}=\frac{T_{eq}}{1+(\lambda/k_{B})S_{q}}\frac{\partial S_{q}}{\partial
V}.$ (17)

We shall see that the Clausius relation modifies due to the above equilibrium
temperature and the equilibrium pressure.

In order to develop the nonextensive thermodynamic relations, we use the
Legendre transformation and the first law of thermodynamics. In [72], the free
energy $F_{q}$, as the Legendre transform structure, in the context of
nonextensive thermodynamics, is defined as

$F_{q}=U_{q}-\frac{1}{k_{B}\beta}S_{q}.$ (18)

In the above equation, the variable in front of $S_{q}$ is the inverse of
Lagrange multiplier $\beta$ which gives the nonphysical temperature. However,
all thermodynamic quantities should be written in physical variables.
Therefore, in [70], Abe et al. proposed the following generalized free energy

$F_{q}=U_{q}-T_{eq}\frac{k_{B}}{\lambda}\ln\left(1+\frac{\lambda}{k_{B}}S_{q}\right),$
(19)

where $\beta^{*}(\beta)$ is introduced which gives the effective equilibrium
temperature $T_{eq}$. In order to define the modified Clausius’ relation, take
the derivative of $F_{q}$ and using the first law of thermodynamics
$dQ_{q}=dU_{q}+P_{eq}dV$, one can write

$\frac{k_{B}}{\lambda}d\ln(1+\frac{\lambda}{k_{B}}S_{q})=\frac{dQ_{q}}{T_{eq}}.$
(20)

which is modified by Clausius’ relation for nonextensive systems. From the
above equations (18) and (19), we can define a new form of entropy and we
denote it by $S_{R}$

$S_{R}=\frac{k_{B}}{\lambda}\ln(1+\frac{\lambda}{k_{B}}S_{q}).$ (21)

Furthermore, the new equilibrium entropy $S_{R}$, by using the composition
rule (12) of $S_{q}$, follows the additive rule, which can be easily shown as

$S_{R12}=S_{R1}+S_{R2}.$ (22)

Interestingly, this new definition of entropy $S_{R}$ happens to be the Rényi
entropy if the Tsallis entropy $S_{q}$ is given.

##  3 Rényi Black Hole Entropy, Temperature, and Mass

In this section, we discuss the application of the above equations (15) and
(21) to the Schwarzschild black hole.

We assume that Bekenstein-Hawking entropy (1) is the Tsallis entropy $S_{q}$
in (21) and write down the corresponding equilibrium entropy $S_{R}$ in terms
of $S_{bh}$ and the equilibrium temperature $T_{eq}$ in terms of $T_{bh}$.

For the case of the Schwarzschild black hole555We do not consider other types
of black holes here, but our formulism is general and can also be applied to
them. For example, one can follow [73]. , we can write the Rényi entropy
$S_{R}$ (8) as

$S_{R}=\frac{1}{\lambda}\ln(1+\pi\lambda r_{h}^{2}),$ (23)

where we have used $k_{B}=1$. The equilibrium temperature $T_{eq}$ (15) as
(cf. formula (19) of Ref. [74])

$T_{eq}=T_{R}=\frac{1}{4\pi r_{h}}+\frac{\lambda r_{h}}{4}.$ (24)

By using the equations (23) and (24), we write the mass $M_{R}$ of the Rényi
black hole by using the relation for the Smarr mass

$M_{R}=2T_{R}S_{R}.$ (25)

Ignoring the higher orders of $\lambda$ by considering small nonextensivity
($\lambda\ll 1$ for $q\approx 1$ ), we can write down equation (25) as

$M_{R}=\frac{r_{h}}{2}+\frac{\lambda\pi r_{h}^{3}}{4}$ (26)

or, explicitly in terms of the Schwarzschild radius $r_{h}=2M$ as

$M_{R}=M+2\pi\lambda M^{3}.$ (27)

This means that the mass of the Rényi black hole $M_{R}$ is the sum of the
mass of the Schwarzschild black hole plus some “environment mass” due to the
extra term in the equation (27). That is, in equation (26), $M_{R}$ is no
longer internal energy due to the additional term. Let us notice that the
equation (26) can be written as

$M_{R}=2T_{bh}S_{bh}+\frac{\lambda\pi r_{h}^{3}}{4}.$ (28)

It is shown in [37, 38] that the nonextensive parameter $\lambda$ can be taken
as a thermodynamic pressure

$P=\frac{3\lambda}{32},$ (29)

and by defining its conjugate variable as the thermodynamic volume
$V=\frac{4\pi}{3}r^{3}_{h}$, we can write a consistent Smarr formula like
equation 666Note that, using equation (25), the equation (30) can be written
as $M=2T_{R}S_{R}-2PV$, where we have used $M=2T_{bh}S_{bh}$ for the
Schwarzschild black hole . [28]

$M_{R}=2T_{bh}S_{bh}+2PV.$ (30)

Thus the Rényi black hole mass, $M_{R}$, should be interpreted as an enthalpy
instead of the internal energy of the black hole, like in the case of an AdS
black hole. In this way, we can write the extended first law of thermodynamics
as

$dM_{R}=T_{bh}dS_{bh}-PdV.$ (31)

Within this context, interesting studies have been investigated in [38, 37,
40] for different black holes. For example, solid, and liquid phase
transition, and latent heat via Rényi extended phase space have been studied,
and black hole heat engines have also been investigated in this scenario. At
least mathematically, we can say that there is an equivalent relation between
the AdS black holes and Rényi black holes.

In [33], the authors analyzed thermodynamic inconsistencies while utilizing
the Rényi black hole entropy with Hawking temperature $T_{bh}$. For instance,
by applying the first law of thermodynamics, $dE_{R}=T_{bh}dS_{R}$, they found

$E_{R}=M-\frac{4\pi\lambda M^{2}}{3},$ (32)

where higher order terms in $\lambda$ are ignored 777Here, parameter $\lambda$
corresponds to the $\alpha$ in the mentioned paper. . The authors contend that
the equation for $E_{R}$ differs from the black hole mass $M$, i.e.,
$E_{R}\neq M$, and that there is no physical explanation for this expression.
Thus, Rényi entropy is not a viable option for black holes. Additionally, they
claim that it conflicts with the principle of energy conservation under the
scenario of spherically symmetric dust shell collapse leading to the formation
of the Schwarzschild black hole. As a result, Rényi entropy cannot be used in
conjunction with the Hawking temperature for black holes. We agree with their
findings. However, there is no valid reason to employ the Hawking temperature
with Rényi entropy. In this article, we present the thermodynamic arguments
for why using Hawking temperature with Rényi entropy is physically unsuitable,
and we analyze the corresponding Rényi temperature that should be used with
Rényi entropy to prevent unphysical scenarios and inconsistencies. We contend
that the assumption that uses the Hawking temperature with the Rényi entropy
is incorrect because, in nonextensive thermodynamics, the Rényi temperature
$T_{R}=T_{eq}$ is the effective temperature associated with the equilibrium
condition (13). The related equilibrium entropy is the Rényi entropy
$S_{eq}=S_{R}$. Therefore, we must utilize the Rényi temperature when
utilizing the Rényi entropy and there is no correspondence between Hawking
temperature and Rényi entropy, so there is no physical justification for
utilizing the Rényi entropy while using the Hawking temperature or vice versa.
It is worth noting that by using, $dE_{R}=T_{R}dS_{R}$, we get $E_{R}=M$,
which gives a consistent thermodynamic relationship between the black hole
energy and mass. Another key reason for not using the Hawking temperature
$T_{bh}$ with the Rényi entropy $S_{R}$ is its inconsistency with the Legendre
structure (18) and (19), which we shall discuss in further detail in one of
the following sections. For example, to describe all thermodynamic quantities
as physical variables, we must utilize physical temperature $T_{eq}=T_{R}$
with $S_{R}$ when defining free energy $F$. See, for instance, equations (18)
and (19). If we use $\beta=1/T_{bh}$ instead of $\beta^{*}=1/T_{eq}$ in $F$,
then $F$ cannot be represented in physical variables, i.e., $\beta$ does not
give the equilibrium condition in this case.

With the preceding arguments, we can conclude that the Rényi temperature and
entropy have valid physical interpretations and that these quantities for a
black hole are well-defined when the black hole is in equilibrium with the
surroundings. This means that $T_{R}$ represents the physical temperature of
the entire system containing a black hole embedded in some surroundings. This
is simply demonstrated by the equation (27), where the first term is the mass
of the Schwarzschild black hole and the second term is due to work done by the
environment. It is worth noting that the higher order terms in $\lambda$ are
disregarded in the equation (27). Additionally, the parameter $\lambda$ is
somewhat related to the cosmological constant, which can be related to the
pressure, giving the same extended thermodynamics for black holes. For
instance, compare equations (2), (3), and (4) with equations (26), (29), and
(31). Note that, in [46], the author used the Padmanabhan thermodynamic
approach, in which the Rényi entropy is specified on the Hubble horizon, to
obtain a term similar to the cosmological constant in the Friedmann equation.
This provides yet another rationale for linking the cosmological constant and
the parameter $\lambda$.

In equation (8), we assumed that $S_{q}=S_{bh}$888In most of the literature
for Rényi black hole entropy, $S_{q}=S_{bh}$ is substituted in $S_{R}$,
because $S_{bh}$ is nonadditive and nonextensive. However, the problem with
this assumption is that, if we rely on Bekenstein’s definition, $S_{bh}$ does
not follow the same composition rule for $S_{q}$.  and also we assume that it
follows the nonadditive composition rule (12). However, in [71], the author
used a unique composition rule by using the definition of Bekenstein entropy
and finding the equilibrium entropy and associated equilibrium temperature. By
following [75, 70], the composition rule for black hole entropy can be written
as

$\frac{S_{bh12}}{k_{B}}=\frac{S_{bh1}}{k_{B}}+\frac{S_{bh2}}{k_{B}}+2\sqrt{\frac{S_{bh1}}{k_{B}}}\sqrt{\frac{S_{bh2}}{k_{B}}},$
(33)

where we consider the case of two black holes, having entropies $S_{bh1}$ and
$S_{bh2}$ and $S_{bh12}$ is the total entropy of composite black hole system
with the inclusion of the long-range force or interactions. By maximizing the
entropy (33) with variation in total mass $\delta
M_{bh12}=\delta(M_{bh1}+M_{bh2})=0$, we have the equilibrium condition

$\frac{\frac{\partial S_{bh1}}{\partial
M_{bh1}}}{k_{B}\sqrt{\frac{S_{bh1}}{k_{B}}}}=\frac{\frac{\partial
S_{bh2}}{\partial M_{bh2}}}{k_{B}\sqrt{\frac{S_{bh2}}{k_{B}}}}=\beta^{*},$
(34)

where the parameter $\beta^{*}$ is defined as

$k_{B}\beta^{*}=\frac{\beta}{\sqrt{\frac{S_{bh}}{k_{B}}}}=\frac{1}{T_{eq}}.$
(35)

Here, we introduced $\beta=\partial S_{bh}/\partial M_{bh}=1/k_{B}T_{bh}$,
which is the usual inverse Hawking temperature. Now we can write $T_{eq}$

$T_{eq}=\frac{\sqrt{\frac{S_{bh}}{k_{B}}}}{k_{B}\beta},$ (36)

and the associated equilibrium entropy can be written as

$S_{eq}=2k_{B}\sqrt{\frac{S_{bh}}{k_{B}}}.$ (37)

Interestingly, the above equilibrium entropy is additive; like in the previous
section, equilibrium entropy $S_{R}$ follows the additive rule for the general
nonextensive case. For the case of the Schwarzschild black hole, the physical
temperature becomes $T_{eq}=1/(4k_{B}\sqrt{\pi})$ and associated equilibrium
entropy reads as $S_{eq}=4\pi k_{B}M$. This means that the equilibrium entropy
is a linear function of the mass of the black hole.

In [75], the same results have been obtained by using the so-called “formal
logarithm” approach [69], and it is shown that, within this approach, pure
isolated black holes are thermodynamically stable against spherically
symmetric perturbations.

##  4 Composition Rule and Equilibrium Temperature for Tsallis-Cirto Black
Hole Entropy

The Legendre transform is significant in classical mechanics, statistical
mechanics, and thermodynamics because it describes how information is coded in
a functional form. It demonstrates how to write a function with the same
information as $F(x)$, but as a function of $dF/dx$. For example, the inverse
temperature $\beta=1/T$ is the conjugate of a system’s total energy $E$.
Despite this, we use the temperature $T$ in the majority of the relationships.
The familiar equation

$F=E-TS$ (38)

which relates the Helmholtz free energy $F$ to the entropy $S$, and it hides
the symmetry between $\beta$ and $E$. However, one can write the duality
between them by writing the dimensionless form of (38). In this way, Gibbs’s
free energy is another example.

By following [59, 60], for a general $d$ dimensional system, the Gibbs free
energy $G$ reads as

$G=U-TS+pV-\mu N,$ (39)

where $T$, $p$, $\mu$, are the temperature, pressure, and chemical potential,
and $U$, $S$, $V$, and $N$ are the internal energy, entropy, volume, and the
number of particles, respectively. Here, $S$, $V$, and $N$ are the extensive
variables scaling with $V=L^{d}$, where $L$ is the linear dimension of
$d$-dimensional system, and the intensive variables $T$, $p$, and $\mu$
scaling with $L^{\theta}$, and finally those variables representing the
energies, $G$ and $U$ scaling with $L^{\epsilon}$. From the above equation, it
follows that

$\epsilon=\theta+d.$ (40)

Schwarzschild ($3+1$)-dimensional black holes have $E=M$ and $M$ scales with
the length scale $L$. Since $\epsilon=1$ for this case, we obtain $\theta=1-d$
from the equation above. Let’s take into account the Bekenstein entropy, which
scales with $L^{2}$. This indicates that the temperature for a Schwarzschild
($3+1$)-dimensional black hole scales with $L^{-1}$, which is precisely true
for Hawking temperature $T_{bh}$. This indicates that the quantities $S_{bh}$
and $T_{bh}$ satisfy the Legendre structure if we think of a black hole as a
two-dimensional object. Additionally, if we consider black holes as three-
dimensional objects based on the aforementioned Legendre structure, the
definitions of entropy and temperature alter for black holes. Tsallis and
Cirto [59, 60] proposed a new type of black hole entropy, and it is defined as
follows:

$S_{T}=k_{B}\left(\frac{S_{bh}}{k_{B}}\right)^{\delta},$ (41)

where $\delta>0$ and its composition rule is given by

$S_{T12}=k_{B}\left[\left(\frac{S_{T1}}{k_{B}}\right)^{1/\delta}+\left(\frac{S_{T2}}{k_{B}}\right)^{1/\delta}\right]^{\delta}.$
(42)

In this context, the $S_{bh}$ is additive, and $S_{T}$ is nonadditive. For
$\delta=3/2$, $S_{T}$ is proportional to the volume for the case of the
Schwarzschild black hole, and so it is extensive. If we consider black hole as
$d=3$ dimensional system, then $S=S_{\delta=3/2}$ and $\theta=-2$, which means
that $T$ must scale with $1/L^{2}$. The corresponding Tsallis-Cirto
temperature can be written by using $S_{T}$ as

$T_{\delta}=\frac{T_{bh}}{\delta}\left(\frac{S_{bh}}{k_{B}}\right)^{1-\delta},$
(43)

which scales with $1/L^{2}$ for $\delta=3/2$, i.e., $T_{\delta}\propto
1/M^{2}$, for the case of Schwarzschild black hole. Now using the equilibrium
condition, we maximize the Tsallis-Cirto black hole entropy $S_{T}$, i.e.,
$\delta S_{T12}=0$ with the assumption that the total energy is fixed, then
the equilibrium condition gives

$\frac{1}{k_{B}}\left(\frac{S_{T1}}{k_{B}}\right)^{\frac{1-\delta}{\delta}}\frac{\partial
S_{T1}}{\partial
U_{1}}=\frac{1}{k_{B}}\left(\frac{S_{T2}}{k_{B}}\right)^{\frac{1-\delta}{\delta}}\frac{\partial
S_{T2}}{\partial U_{2}}=\beta^{*},$ (44)

which means that the equilibrium temperature for this case can be written as

$T_{eq}=\frac{1}{k_{B}\beta^{*}}=T_{\delta}\left(\frac{S_{T}}{k_{B}}\right)^{\frac{\delta-1}{\delta}},$
(45)

and corresponding equilibrium entropy can be written as

$S_{eq}=k_{B}\delta\left(\frac{S_{T}}{k_{B}}\right)^{1/\delta}.$ (46)

Interestingly, using the values of $S_{T}$ (41) and $T_{\delta}$ (43) in above
equations (45) and (46), we get $T_{eq}=T_{bh}/\delta$ and corresponding
equilibrium entropy would be $S_{eq}=\delta S_{bh}$. Again, in the context of
the above composition rule (42), the equilibrium entropy for this case is also
additive. Note that, now the definitions of equilibrium entropy $S_{eq}=\delta
S_{bh}$ and equilibrium temperature $T_{eq}=T_{bh}/\delta$ are defined in
terms of $S_{T}$ and $T_{\delta}$. In this regard, we can say that the
Bekenstein-Hawking entropy and Hawking temperatures are the equilibrium
entropy and equilibrium temperature for the nonextensive Tsallis-Cirto setup.

Let us revisit the Legendre structure in this situation. Similarly to the
situation of $S_{R}$, in [33], the authors utilized the argument against the
Hawking temperature $T_{bh}$ (1) associated usually with Tsallis-Cirto black
hole entropy (41) in numerous applications. In this context, they analyze that
$E_{T}\neq M$ by applying the relation $dE_{T}=T_{bh}dS_{T}$, and therefore
$T_{bh}$ is not an appropriate choice to employ with $S_{T}$. The authors
provided no reason for using $T_{bh}$ with $S_{T}$. This is merely an
assumption; we will always obtain nonphysical results if we make inappropriate
assumptions because the relationship $dE=TdS$ between temperature and entropy
is required for a well-defined Legendre structure. For example, $T_{bh}$ and
$S_{T}$ cannot be used in the thermodynamic potential $G$ because both are
incompatible. To avoid unphysical outcomes, one must utilize equivalent
compatible physical quantities, such as $T_{\delta}$ with $S_{T}$ with
$dE_{T}=T_{\delta}dS_{T}$.

Formally, Tsallis-Cirto entropy as given by the formula (41) together with
(1), is the same as the Barrow entropy [61] which is defined as

$S_{B}=k_{B}\left(\frac{A}{A_{Pl}}\right)^{1+\frac{\Delta}{2}},$ (47)

where $A_{Pl}$ is the Planck area, and $0\leq\Delta\leq 1$. While comparing
both definitions, we can see that [76]

$1+\frac{\Delta}{2}=\delta,$ (48)

for both formulas to be (up to a factor) the same. The extensive limit of the
nonextensive Barrow entropy is given for $\Delta=1$, which corresponds to
$\delta=3/2$ (extensive) limit of the Tsallis-Cirto entropy.

However, Barrow’s entropy comes from purely geometrical or rather fully non-
thermodynamical motivation. Shortly, the idea is to replace a black hole
smooth spherical horizon with the fractal structure of spheres attached
infinitely to the spherical horizon forming the so-called sphere flake,
characterized by a fractal dimension $D_{f}$ falling into the interval $2\leq
D_{f}\leq 3$. This leads to an effective horizon sphere radius to be

$r_{eff}=r^{1+\Delta/2},$ (49)

where $r$ is the radius of the non-fractal horizon. The horizon area is then
modified accordingly

$A_{eff}=4\pi r_{eff}^{2},$ (50)

and so is the (area) entropy.

Despite that, it seems that Tsallis-Cirto thermodynamics can fully be applied
to the Barrow entropy within the range of Tsallis-Cirto nonextensivity
parameter $1<\delta<3/2$. This also means that the equilibrium temperature for
Barrow entropy can also be defined, and it falls into the same formula (up to
some factors) as for the Tsallis-Cirto entropy as given by (43). Barrow
entropy has recently been used in many cosmological horizon applications
claiming to serve as holographic dark energy [43, 77, 78, 53, 79, 52].

##  5 Conclusions

We have explored several aspects of the nonextensive thermodynamics of black
holes. In particular, by maximizing various nonextensive entropies defined on
the event horizon, we have studied the equilibrium temperature for a
Schwarzschild black hole and obtained the equilibrium conditions in the
nonextensive setting. We have come to the conclusion that there is always an
equilibrium temperature in the nonextensive setup which is different from the
absolute temperature and corresponds to an additive equilibrium entropy that
is different from the nonextensive one.

The primary purpose of our study has been to determine whether the Hawking
temperature was appropriate for black holes and other cosmological
applications in the nonextensive scenario. In this respect, we have shown that
Hawking temperature is not a consistent thermodynamic quantity to take into
account while studying the nonextensive entropy of black holes and
cosmological horizons. For instance, we have shown that the Legendre structure
is not valid when one associates the Hawking temperature with the Rényi black
hole entropy and Tsallis-Cirto black hole entropy. Furthermore, we have found
in the Tsallis nonextensive setup that the Rényi temperature was the
equilibrium temperature and Rényi entropy was the equilibrium entropy for
black holes.

In the nonextensive setup, the assumption of Bekenstein entropy as Tsallis
entropy is unclear. The nonextensive nature of Bekenstein entropy provides the
basis of this supposition. Bekenstein entropy, on the other hand, follows a
specific nonadditive composition rule based on the entropy-area relation
rather than the generic nonextensive composition rule. In this context, we
have explored the equilibrium temperature by maximizing the Bekenstein
entropy, which is simply a constant independent of the mass of the black hole,
and the associated equilibrium entropy in this case is proportional to the
mass of the black hole.

Moreover, by maximizing the Tsallis-Cirto black hole entropy, we have
investigated the equilibrium temperature and have demonstrated that the
Hawking temperature is the equilibrium temperature, and the Bekenstein entropy
is the corresponding equilibrium entropy for such a case. A similar result is
true for the case of Barrow entropy, too.

Numerous applications of non-extensive entropies in gravitation and cosmology
have been made recently, yet there are still many fascinating concerns that
remain unresolved. With nonextensive entropies, in particular, the Hawking
temperature has been often utilized despite being inconsistent with
nonextensive entropies except the standard Bekenstein entropy. Therefore,
using the consistent temperature corresponding to each nonextensive entropy,
we explored the consistent nonextensive thermodynamic quantities for black
holes. In the future, when we get to these problems, it would be intriguing to
look into the nonextensive thermodynamics of cosmological horizons with
consistent nonextensive thermodynamic quantities. Similarly, it would be
interesting to investigate the second law of thermodynamics and Bekenstein
bound within this framework. Moreover, modified cosmology from a thermodynamic
perspective with consistent thermodynamic quantities would be more
interesting.

### Acknowledgements

The work of I.C. and M.P.D. was supported by the Polish National Science
Centre grant No. DEC-2020/39/O/ST2/02323.

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