# Generalized Uncertainty Principle Impact on Nonextensive Black Hole
Thermodynamics

Ilim Çimdiker<EMAIL_ADDRESS>Institute of Physics, University
of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland  Mariusz P. Da̧browski
<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
Gohar<EMAIL_ADDRESS>Institute of Physics, University of Szczecin,
Wielkopolska 15, 70-451 Szczecin, Poland

(August 30, 2024)

###### Abstract

The effect of the generalized uncertainty principle (GUP) on nonextensive
thermodynamics applied to black holes, as well as the sparsity of the
radiation at different temperatures associated with each nonextensive entropy,
is investigated. We examine the Rényi, Tsallis-Cirto, Kaniadakis, Sharma
Mittal, and Barrow entropies, temperatures, and heat capacities and show that,
in each case, due to GUP corrections, the temperature and entropy have finite
values, implying that the final state of the black hole is a remnant at the
end of the evaporation process and that the sparsity of the radiation for
massless bosons at each temperature depends on the mass of the black hole. We
also find that GUP reduces the value of the sparsity profile for each case as
compared to the sparsity parameter at Hawking temperature, which is always
constant throughout the evaporation.

##  I Introduction

Black holes emit radiation due to the Hawking evaporation process, and
therefore, there is an established concept of Hawking temperature Hawking
(1974) and Bekenstein entropy Bekenstein (1973) connected with the black hole
horizon. The black hole evaporation process operates within the purview of
quantum field theory, and one of its more intriguing aspects may be that it
appears to indicate a non-unitary evolution, which gives rise to the well-
known issue of the information loss paradox Hawking (1976a); Chen  _et al._
(2015); Unruh and Wald (2017). Black holes behave like thermodynamic objects,
and the laws of black hole thermodynamics Bardeen  _et al._ (1973); Gibbons
and Perry (1978); Hawking and Page (1983); Hawking (1976b, 1975) are analogous
to the conventional thermodynamic laws. The thermodynamics of black holes have
been extensively studied and used in a variety of cosmological and
gravitational applications Jacobson (1995); Verlinde (2011); Padmanabhan
(2010); Kubiznak and Mann (2015); Cvetic  _et al._ (2011); Caldarelli  _et
al._ (2000); Cai and Kim (2005); Davies (1977); Dolan (2012); Easson  _et al._
(2011).

Entropy measures how difficult it is for an outside observer to get
information about the underlying structure of the system. This is a clear
reflection of the macroscopic features that result from the quantum
statistical mechanics that govern the behavior of quantum microstates. For the
case of black holes, there is no definition of Bekenstein entropy in quantum
statistical mechanics and it only relies on Hawking’s area theorem Hawking
(1971), therefore, it would be required to have a complete theory of quantum
gravity in order to fully comprehend the origin of this entropy and the nature
of microstates in the case of black holes. In its absence, we rely on the
definition of Bekenstein entropy for black holes. For the case of a
Schwarzschild black hole with mass $M$, the Hawking temperature $T_{H}$ and
Bekenstein entropy $S_{B}$ are given by Hawking (1974); Bekenstein (1973)

$T_{H}=\frac{\hbar\kappa}{2\pi k_{B}c}\leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
S_{B}=\frac{k_{B}c^{3}A}{4G\hbar}\leavevmode\nobreak\ \leavevmode\nobreak\ ,$
(1)

where $\hbar$, $G$, $k_{B}$, and $c$ are the reduced Planck constant, the
Newton gravitational constant, the Boltzmann constant, and the speed of light,
respectively. The area $A$ of the event horizon is defined as $A=4\pi
r_{h}^{2}$ in the above equation (1), where $r_{h}=2GM/c^{2}$ is the
Schwarzschild radius and $\kappa=c^{4}/4\pi GM$ is the surface gravity defined
on the event horizon of the Schwarzschild black hole.

Gibbs statistical mechanics is based on two key hypotheses: that entropy is
extensive and that internal energy and entropy follow the additive composition
rule. All thermodynamic relations in Gibbs statistical mechanics are defined
in light of these presumptions. It is very important to differentiate
extensivity and additivity of a thermodynamic quantity in general (for more
comprehensive discussion, see Refs. Landsberg (1999); Swendsen (2011);
Mannaerts (2014). Assume two independent systems $A$ and $B$ with an ensemble
of configurational possibilities $\Omega_{A}$ and $\Omega_{B}$ and
corresponding probabilities $P_{A}$ and $P_{B}$. Consider $AUB$ now, with
$P_{AUB}$ being the probability and $\Omega_{AUB}$ being the set of
possibilities. Because of the systems’ independence, $P_{AUB}=P_{A}P_{B}$.
Therefore, if $S(A+B)\equiv S(P_{A}P_{B})=S(P_{A})+S(P_{B})\equiv S(A)+S(B)$,
then an entropy functional $S(P)$ is said to be additive. In order to define
extensivity, we will use Tsallis’ definition of extensive entropy, which
states that if a system’s total number of microstates, $\Omega$, is
proportional to its number of particles or degrees of freedom, the entropy is
extensive. For instance, the Gibbs entropy is defined as
$S_{G}(N)=k_{B}\ln\Omega(N)\propto N$, where $N$ is the total number of
particles or degrees of freedom in the system. Keep in mind that extensive
entropy can be nonadditive. In Gibbs thermodynamics, entropy is defined as
extensive because it scales with the size of the system. This definition does
not capture its full significance, and is not stated with full mathematical
rigour: what does it mean to ‘scale’? What is meant by ‘size’ ? Is it the
volume? Mole number? Both? In order to understand the definition of extensive
variables more clearly, we define a function $f$, the fundamental relation of
thermodynamic variables ($X_{0}$, $X_{1}$,$X_{2}$,…,$X_{k}$) such that
$X_{0}=f(X_{1},X_{2},...,X_{k})$. Here, $f$ is homogeneous first order
function of $X_{1}$,$X_{2}$,…,$X_{k}$ when
$f(aX_{1},aX_{2},...,aX_{k})=af(X_{1},X_{2},...,X_{k})$ for every positive
real numbers $a$ for all $X_{1}$, $X_{2}$, …$X_{k}$. The thermodynamic
variables $X_{i}$ can be the energy $U$, entropy $S$ and mole number $N$ and
expressing $f$ in differential form will give the first law of thermodynamics.
For example, in Gibbs thermodynamics, the fundamental relation $f$ for the
entropy $S$ can be written as $S=f(U,V,N)$ for an ideal case and and
$f(aU,aV,aN)=af(U,V,N)$, hence $S$ is extensive. In terms of the scaling
symmetry of the fundamental relation, the geometric framework provides a
precise way of defining what extensive variables are. In a nutshell, we will
say that a set of thermodynamic variables is "extensive" when the first-order
homogeneous property is imposed on the fundamental relation. This way, we
avoid ambiguity in the word ’size,’ as well as claims that volume and mole
number are ’obviously’ extensive, as seen frequently in discussions of
extensivity.

Nonextensive statistical mechanics, such as Tsallis nonextensive statistical
mechanics Tsallis (1988); Tsallis  _et al._ (1998); Tsallis (2009); Abe  _et
al._ (2001); Abe and Rajagopal (2001); Abe (2001); Biró and Ván (2011);
Nauenberg (2003); Biró and Ván (2011); Parvan and Biro (2005), is the outcome
of removing the assumption of extensivity. The assumption of the extensive
nature of entropy is connected to ignoring the long-range forces between
thermodynamic sub-systems. Since the size of the system exceeds the range of
the interaction between the system’s components, Gibbs thermodynamics ignores
these forces. Because of this, the total entropy of a composite system equals
the sum of the entropies of the individual subsystems and entropy grows with
the size of the system. However, long-range forces are important in various
unique thermodynamic systems. For instance, if we think of a black hole as a
$(3+1)$ dimensional object, it is vital to note that Bekenstein entropy scales
with the area and is thus regarded as a nonextensive quantity Tsallis and
Cirto (2013); Biró and Czinner (2013); Czinner (2015); Czinner and Iguchi
(2017a, 2016, b); Tsallis (2019). Furthermore, because of the area scaling,
Bekenstein entropy is nonadditive and follows a nonadditive composition rule
$S_{12}=S_{1}+S_{2}+2\sqrt{S_{1}}\sqrt{S_{2}}$ (see e.g. Alonso-Serrano  _et
al._ (2021)), whereas Gibbs statistical mechanics or thermodynamics is based
on the extensive and additive properties of the entropy. Therefore, Gibbs
thermodynamics or statistical mechanics may not be the appropriate choice for
studying the thermodynamics of black holes. In order to understand the
nonextensive and nonadditive nature of Bekenstein entropy, several extensions
Tsallis (1988); Rényi (1959); Sharma and Mittal (1977, ); Kaniadakis (2002,
2005); Barrow (2020) of standard Gibbs thermodynamics have been applied to
black holes and cosmological horizons Nojiri  _et al._ (2021, 2022a, 2022b,
2022c); Promsiri  _et al._ (2020, 2021); Tannukij  _et al._ (2020);
Nakarachinda  _et al._ (2021); Çimdiker  _et al._ (2022); Promsiri  _et al._
(2022); Nakarachinda  _et al._ (2022); Saridakis (2020); Dabrowski and Salzano
(2020); Nojiri  _et al._ (2022a, d); Komatsu (2017); Komatsu and Kimura
(2016); Nunes  _et al._ (2016); Liu (2022); Majhi (2017); Luciano and Blasone
(2021); Di Gennaro and Ong (2022); Di Gennaro  _et al._ (2022); Asghari and
Sheykhi (2022); Abreu and Neto (2022); Sayahian Jahromi  _et al._ (2018);
Drepanou  _et al._ (2022). One of the main proposals is the Tsallis-Cirto’s
black hole entropy definition Tsallis and Cirto (2013), which makes the black
entropy extensive and compatible with the Legendre structure. Rényi entropy
Rényi (1959), being a measure of entanglement, is another definition of
entropy applied to black holes and cosmological horizons which is
nonextensive, but additive (by assumption). There have been some other
nonextensive forms of entropy suggested such as the Sharma-Mittal entropy
Sharma and Mittal (1977, ) as a generalization of Rényi entropy, the
Kaniadakis entropy Kaniadakis (2002) which takes inspiration from Lorentz
group transformations and the Barrow entropy Barrow (2020) which is based on a
hypothetical fractal structure of black hole horizon as a result of quantum
fluctuations.

Due to the prevalence of quantum gravity effects, it is anticipated that the
semiclassical technique would fail during the last phases of Hawking
evaporation. There is currently no satisfactory theory of quantum gravity that
enables us to completely explain that regime, despite the development of
several quite diverse proposals Carlip (2001); Konishi  _et al._ (1990); Adler
and Santiago (1999); Rovelli (1996); Meissner (2004); Scardigli (1999);
Hossenfelder (2013). Investigating the phenomenological consequences of an
underlying theory of quantum gravity is one technique to explore the quantum
gravity effects at those scales. The generalized uncertainty principle (GUP)
Maggiore (1993); Scardigli (1999); Giddings (1992); Hossenfelder (2013) is one
approach that has the benefit of being sufficiently generic to be compatible
with several quantum gravity theories. The Bekenstein entropy and Hawking
temperature of a black hole in its last phases of evaporation are modified
within this framework Adler and Santiago (1999). Because of these
modifications, black holes do not entirely evaporate during the evaporation
process, and the final state of the black hole is a remnant of the order of
Planck mass

Sparsity Page (1976a, b, 1977); Schuster (2018); Gray  _et al._ (2016);
Schuster (2021); Paul and Majhi (2017); Alonso-Serrano  _et al._ (2018a, b);
Ong (2018a); Alonso-Serrano  _et al._ (2021); Feng  _et al._ (2020) is an
important feature of Hawking radiation. It is defined as the average time
between the emission of successive quanta over the timescales set by the
energies of the emitted quanta. It was shown that Hawking radiation is very
sparse during the black hole evaporation process Gray  _et al._ (2016), which
is one of the key characteristics that distinguish it from black-body
radiation. However, it has been found that when GUP corrections are
incorporated Alonso-Serrano  _et al._ (2018a, b); Ong (2018a), the sparsity
decreases toward the late stages of evaporation. When nonextensivity is
considered in the context of Rényi temperature Alonso-Serrano  _et al._
(2021), the Rényi radiation is initially not sparse, but as evaporation
progresses, it begins to become sparse and eventually approaches the case of
Hawking radiation.

In this paper, we are interested in exploring the GUP modifications to the
nonextensive entropies and corresponding thermodynamic quantities in Rényi,
Tsallis-Cirto, Sharma-Mittal, Kaniadakis, and Barrow nonextensive statistics.
Furthermore, the sparsity of the radiation is analyzed at different
temperatures corresponding to different nonextensive entropies.

The following is the outline of the paper. In Sec. II, we introduce the notion
of GUP and apply it to the case of standard thermodynamic black hole
quantities. In Sec. III, we introduce nonextensive entropies and accompanying
nonextensive thermodynamic quantities, as well as GUP modifications to
nonextensive black hole thermodynamics. Finally, in Sec. IV, we summarize and
discuss our findings.

##  II GUP and Black Hole Thermodynamics

###  II.1 Generalized Uncertainty Principle

One common aspect of several quantum gravity theories is that they all predict
a minimum measurable length Hossenfelder (2013); Amati  _et al._ (1989). For
example, the notion of minimal length is defined in string theory as the
string length Kempf  _et al._ (1995); Konishi  _et al._ (1990), in loop
quantum gravity Rovelli (1996) it is the expectation value of the length
operator, and this notion can also be developed by the phenomenological
aspects coming from black hole physics Hossenfelder (2013). Because of the
appearance of another minimum length at the Planck scale in various quantum
gravity approaches, it has been proposed that the Heisenberg Uncertainty
Principle (HUP)

$\Delta x_{0}\Delta p\geq\hbar,\hskip 5.69046pt{\rm or}\hskip 5.69046pt\Delta
x_{0}\sim\frac{\hbar}{\Delta p}$ (2)

where $\Delta x_{0}$ and $\Delta p$ are position and momentum uncertainties
can be modified when gravitational interaction is introduced. The simplest
argument for the modification of HUP within the framework of Newtonian theory
is that there is a gravitational acceleration $\vec{a}$ of an electron due to
a photon of mass $E/c^{2}$ Adler and Santiago (1999), where $E$ is the photon
energy and $r$ is the photon-electron distance, which reads

$\vec{a}=\ddot{\vec{r}}=-\frac{G(E/c^{2})}{r^{2}}\frac{\vec{r}}{r},$ (3)

and the interaction takes place in a characteristic region of length $L\sim r$
and in characteristic time $t\sim L/c$. Then, the velocity acquired by an
electron $\Delta v$ is

$\Delta v\sim\frac{GE}{c^{2}r^{2}}\frac{L}{c},$ (4)

and the (extra due to gravity) distance $\Delta x_{1}$ it is shifted reads

$\hskip 14.22636pt\Delta
x_{1}\sim\frac{GE}{c^{2}r^{2}}\frac{L^{2}}{c^{2}}\sim\frac{G\Delta
p}{c^{3}}=\frac{c\Delta p}{4F_{pl}}=l_{p}^{2}\frac{\Delta p}{\hbar},$ (5)

where $l_{p}=\sqrt{G\hbar/c^{3}}$ is the Planck length, and $F_{pl}=c^{4}/4G$
is the Planck force (often called the maximum force in the context of general
relativity) Schiller (2005); Barrow and Gibbons (2014); Dabrowski and Gohar
(2015); Ong (2018b). Extra uncertainty (5) adds to the standard HUP
uncertainty of position $\Delta x_{0}$ as in (2) giving

$\Delta x=\Delta x_{0}+\Delta x_{1}\sim\frac{\hbar}{\Delta
p}+l_{p}^{2}\frac{\Delta p}{\hbar},$ (6)

leading to the generalized uncertainty principle (GUP)

$\Delta x\Delta p\geq\hbar\left(1+\frac{l^{2}_{p}}{\hbar^{2}}(\Delta
p)^{2}\right).$ (7)

Taking an algebraic point of view, GUP can be derived from the deformed
commutation relation between the position operator $\hat{x}$ and the momentum
operator $\hat{p}$ such that

$\left[\hat{x},\hat{p}\right]=i\hbar f(\hat{p}),$ (8)

where $f(\hat{p})$ is a general function of momentum operator $\hat{p}$ and
there exist different proposed functions for $f(\hat{p})$. In order to make
the function $f(\hat{p})$ compatible with (7), following the literature, we
choose

$f(\hat{p})=1+\alpha\frac{l_{p}^{2}}{\hbar^{2}}\hat{p}^{2},$ (9)

where we the introduce GUP parameter $\alpha$ – a dimensionless parameter
predicted to be of order of unity, but there are different (mostly upper)
bounds on it from different experiments and observations Gao and Zhan (2016);
Feng  _et al._ (2017); Bosso  _et al._ (2018); Gao  _et al._ (2017); Giardino
and Salzano (2021). By introducing $\alpha$, the equation (10), now, reads as

$\Delta x\Delta p\geq\hbar\left[1+\alpha\frac{l^{2}_{p}}{\hbar^{2}}(\Delta
p)^{2}\right].$ (10)

####  II.1.1 GUP Modified Hawking Temperature and Bekenstein Entropy

An interesting application of (10) to black hole physics is the modification
to the Hawking temperature, which can be derived by solving it for $\Delta p$,
which gives

$\displaystyle\Delta p=\Delta x\frac{\hbar}{\alpha
l^{2}_{p}}\left[1\pm\sqrt{1-\frac{\alpha l^{2}_{p}}{(\Delta x)^{2}}}\right].$
(11)

We consider the ${}^{\prime\prime}+^{\prime\prime}$ sign in (11) as
$\alpha\rightarrow 0$ limit yields the standard Heisenberg uncertainty
principle, whereas the negative sign does not. Considering the minimum
position uncertainty near the event horizon of the Schwarzschild black hole as
$\Delta x=2l_{p}=4GM/c^{2}$, where $l_{p}$ is taken as the Schwarzschild
radius $r_{h}$, the GUP modified Hawking temperature $T_{GUP}$ reads

$\displaystyle T_{GUP}=\frac{m_{p}^{2}c^{2}}{8\pi
k_{B}M}\left[\frac{4}{2+\sqrt{4-\alpha\frac{m_{p}^{2}}{M^{2}}}}\right].$ (12)

By introducing a correction term due to GUP, $\mathcal{K}(\alpha,M)$,
$T_{GUP}$ can be written in terms of $T_{H}$ and $\mathcal{K}$, such that

$T_{GUP}=T_{H}(M)\mathcal{K}(\alpha,M),$ (13)

where the GUP correction term is defined as

$\mathcal{K}(\alpha,M)=\frac{4}{2+\sqrt{4-\alpha\frac{m_{p}^{2}}{M^{2}}}}.$
(14)

This provides us with a more compact form of $T_{GUP}$, which will be used in
the next sections for GUP modifications to the thermodynamic quantities. Note
that we consider the case where $M^{2}\geq\alpha m_{p}^{2}/4$ to make the
parameter $\mathcal{K}$ real valued function.

Using the Clausius relation, the GUP modified Bekenstein entropy $S_{GUP}$ in
terms of $S_{B}$ and the correction term $\mathcal{K}(\alpha,M)$ can be
written as

$\displaystyle S_{GUP}=\frac{S_{B}}{\mathcal{K}}-\frac{\alpha\pi
k_{B}}{2}\ln{\left[\frac{4M}{m_{0}\mathcal{K}}\right]},$ (15)

where $m_{0}$ is a dimensionful constant of unit mass, which is introduced in
order to make the logarithm dimensionless. In the limit $\alpha\rightarrow 0$,
the correction term $\mathcal{K}$ goes to one, and hence $T_{GUP}$ and
$S_{GUP}$ reduce to $T_{H}$ and $S_{B}$. The plots of (12) and (15) are given
in Figs. 1 and 2. Note that all the plots in the paper, unless explicitly
stated, are given in natural units $\hbar=c=G=1$ and also with the GUP
parameter $\alpha=1$.

Figure 1: Temperature vs mass for the Hawking temperature $T_{H}$ and the GUP
corrected temperature with positive and negative values of $\alpha$. Threshold
with positive $\alpha$ for mass lies at the remnant mass
$M_{r}^{2}=(\alpha/4)m_{p}^{2}$ (cf. formula (16)). Figure 2: Entropy vs mass
for the Hawking temperature and GUP corrected temperatures with positive and
negative values of $\alpha$. The threshold for mass lies at the remnant mass
given by $M_{r}^{2}=(\alpha/4)m_{p}^{2}$.

For positive values of $\alpha$, the black hole evaporation stops when the
mass of the black hole reaches some critical value of mass

$M_{r}=\frac{\sqrt{\alpha}m_{p}}{2}=\frac{l_{p}\sqrt{\alpha}}{2c^{2}}F_{pl},$
(16)

which is called the black hole remnant mass and we introduce the Planck force
$F_{pl}=c^{4}/G$ in above equation. Therefore, we can say that the final state
of the black hole evaporation is a remnant having the mass $M_{r}$. In fact,
without a well-defined quantum gravity theory, we cannot predict what happens
if the mass of a black hole is smaller than this critical value. For the
critical mass value $M_{r}$, the formulas (12) and (15) for $T_{GUP}$ and
$S_{GUP}$, give the temperature $T_{r}$ and the entropy $S_{r}$ for the
remnant as Alonso-Serrano  _et al._ (2021)

$T_{r}=\frac{m_{p}c^{2}}{2\pi k_{B}\sqrt{\alpha}},\leavevmode\nobreak\
S_{r}=\frac{\pi\alpha
k_{B}}{2}\left[1-\ln{\left(\frac{\sqrt{\alpha}m_{p}}{m_{0}}\right)}\right],$
(17)

provided that $\alpha>0$. For $\alpha<0$ in (14), we have a smooth correction
function defined for all black hole mass values. In this case, the black hole
continues to radiate slowly and yields an infinite lifetime Ong (2018a). When
$M$ approaches zero, interestingly, the temperature is still finite, and for
this case, in Jizba  _et al._ (2010), it is referred to as a remnant with zero
rest mass.

####  II.1.2 GUP Modified Heat Capacity

In order to investigate the GUP modifications to the heat capacity of a black
hole with mass $M$, we use the definition of heat capacity $C$, which reads

$\displaystyle C=-\frac{S^{\prime 2}(M)}{S^{\prime\prime}(M)},$ (18)

where $S$ is the black hole entropy and prime and double prime denote the
first and second derivative with respect to the mass $M$. For the case of
Schwarzschild black hole, we have (denoting $C$ as $C_{Sc}$)

$\displaystyle C_{Sc}=-8\pi k_{B}\frac{M^{2}}{m_{p}^{2}},$ (19)

and we can see that it is negative for all mass values. This means that the
Schwarzschild black hole is thermodynamically unstable. In order to introduce
GUP corrections, we introduce the quantity

$\beta_{GUP}=\frac{1}{k_{B}T_{GUP}},$ (20)

which after using (12) gives

$\displaystyle\frac{S_{GUP}^{\prime}(M)}{k_{B}c^{2}}=\beta_{GUP}=\frac{\beta}{\mathcal{K}}\;,$
(21)

where $\beta=1/k_{B}T_{H}$ is the inverse Hawking temperature. Differentiating
$\beta_{GUP}$ once more, and using equations (18) and (21), we obtain the GUP
modified heat capacity $C_{GUP}$, which can be written as (cf. Fig. 3)

$\displaystyle
C_{GUP}=C_{Sc}\left[\frac{2-\mathcal{K}}{\mathcal{K}^{2}}\right].$ (22)

This means that the GUP corrections still yield a negative heat capacity for
$M>M_{r}$, and when the black hole mass approaches the critical mass $M_{r}$,
we have $\mathcal{K}=2$ and interestingly, we get the zero heat capacity for
the remnant. In such a case, from the thermodynamic point of view, a small
amount of heat would then increase the temperature of the remnant by an
infinite amount.

Figure 3: Specific heat capacity of the Hawking radiation for GUP corrected
black holes. For positive $\alpha$, there is a remnant with zero heat
capacity.

####  II.1.3 GUP Modified Sparsity of Hawking Radiation

One of the most important aspects of Hawking radiation is that it is extremely
sparse as compared to black-body radiation. The sparsity can be defined by
using the parameter $\eta$ Gray  _et al._ (2016); Alonso-Serrano  _et al._
(2018a, 2021),

$\displaystyle\eta=\frac{C}{g}\left(\frac{\lambda^{2}_{t}}{A_{eff}}\right),$
(23)

where $C$ is a dimensionless constant associated with different physical cases
Gray  _et al._ (2016), $g$ is the spin degeneracy factor of the particle,
$\lambda_{t}=2\pi\hbar c/k_{B}T$ is the thermal wavelength in terms of the
temperature $T$ and

$A_{eff}=27A/4$ (24)

is the effective area with $A$ being the horizon area for the case of black
holes Page (1976a); Gray  _et al._ (2016). For the Schwarzschild black hole,
one can find the thermal wavelength $\lambda_{t}$ by taking
$T=T_{H}=1/k_{B}\beta$ as

$\displaystyle\lambda_{t}=\frac{2\pi\hbar c}{k_{B}T_{H}}=2\pi\hbar c\beta,$
(25)

and the sparsity profile for massless bosons in the Hawking process yields
111Here the sparsity profile $\eta_{H}$ does not represent the actual value
for sparsity $\eta$ which will include spin degeneracy factor $g$ and $C$ a
dimensionless constant which depends on the chosen time scale. Here we look
for the qualitative behaviour of sparsity with respect to mass which depends
on sparsity profile through temperature and area of the black hole.

$\displaystyle\frac{\lambda^{2}_{t}}{A_{eff}}=\eta_{H}=\frac{64\pi^{3}}{27}\approx
73.38,$ (26)

which does not depend on mass of the black hole. Note that for classical black
body radiation, the value of $\eta$ is less than one. This implies that the
sparsity parameter clearly differentiates the Hawking radiation from classical
radiation. One can obtain the GUP effects on the sparsity by replacing the
Hawking temperature with the GUP corrected temperature $T_{GUP}$ given by (12)
Alonso-Serrano  _et al._ (2018a). However, it is assumed that GUP also
modifies the black hole horizon area Alonso-Serrano  _et al._ (2018a, 2021).
Thus, it is logical to take the effective area that GUP modifies. In fact, the
GUP modifications to $A$ can be derived from the equation (15) by writing it
as

$S_{GUP}=\frac{k_{B}c^{3}A_{GUP}}{4\hbar G},$ (27)

where the GUP modified area $A_{GUP}$ reads

$A_{GUP}=\frac{A}{\mathcal{K}}-\alpha\pi
l_{p}^{2}\ln\left(\frac{16A}{A_{0}\mathcal{K}^{2}}\right),$ (28)

and $A_{0}=16\pi m_{0}^{2}G^{2}/c^{4}$ is a constant having the dimension of
area. Note that in Alonso-Serrano  _et al._ (2021), corrections are only in
the first order of $\alpha$, while in the above equation (28) the area is
corrected to all orders in $\alpha$. Sparsity depends on the crossectional
area of the body at the ray optics limit and the corresponding temperature of
the body, which directly depends on the horizon area and the entropy
associated with the body, respectively. Thus we heuristically obtain the GUP
corrected sparsity profile by replacing $T$ by $T_{GUP}$ and $A$ by $A_{GUP}$
in the expressions for $A_{eff}$ in (24) and for $\lambda_{t}$ in (25). It
then reads

$\displaystyle\eta_{GUP}=\frac{\eta_{H}}{\mathcal{K}^{2}}\left[\frac{A}{A_{GUP}}\right].$
(29)

Interestingly, GUP modified sparsity profile $\eta_{GUP}$, depends on the mass
of the black hole and the GUP parameter $\alpha$. For the negative values of
$\alpha$, the sparsity profile increases as $M$ goes to zero. For the positive
values of $\alpha$, the sparsity parameter decreases below the values of
sparsity for the Hawking radiation until it reaches the critical mass $M_{r}$.

Figure 4: Sparsity profile of Hawking vs GUP corrected black holes in natural
units. For positive values of $\alpha$, we observe that sparsity decreases
when a black hole is near the final evaporation state.

In Fig. 4, we can see that the GUP corrected sparsity profile is not a
constant and it increases first before $M$ approaches $M_{r}$ for $\alpha>0$
and then it decreases to finite value when $M$ approaches to $M_{r}$. For the
case of $\alpha<0$, first, it decreases, and then it goes to plus infinity
when $M$ approaches zero. It is due to the fact that $A/A_{GUP}>1$ for
$\alpha>0$ and $\eta_{H}/\mathcal{K}^{2}$ turns back the sparsity profile from
a maximum value to a constant value, which is less than $\eta_{H}$. Therefore,
we can clearly see the effects of GUP on sparsity due to $T_{GUP}$ and
$A_{GUP}$ as depicted in Fig. 4. Similarly, $A/A_{GUP}<1$ for $\alpha<o$ and
$\mathcal{K}$ goes to zero when $M$ approaches zero, therefore, sparsity
decreases first, and then it goes to infinity. Note that in Ong (2018a), the
GUP corrected area is not taken into account, therefore, there is no bump in
the sparsity profile.

##  III GUP and Nonextensive Black Hole Thermodynamics

###  III.1 Tsallis Nonextensive Entropy

Entropy plays a significant role in Gibbs thermodynamics or statistical
mechanics. It is extensive and adheres to the additive composition rule.
However, Gibbs statistical mechanics ignores long-range forces. Hence, there
are some physical systems for which Gibbs thermodynamics cannot be the
appropriate choice to apply Tsallis (2009) since they are subject to such
forces. Important examples are some self-gravitating systems such as black
holes, for which the forces are long-distance and play some significant role.
For that reason Constantino Tsallis in Refs. Tsallis (1988, 2009) generalized
the conventional Gibbs entropy for nonextensive systems in order to encompass
and address this issue. Tsallis entropy $\mathcal{S}_{\mathcal{T}}$ was one of
the earliest proposals to extend Gibbs entropy and the suggested new form of
it reads

$\mathcal{S}_{\mathcal{T}}=-k_{B}\sum_{i}[p(i)]^{q}\ln_{q}p(i),$ (30)

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

$\ln_{q}p=\frac{p^{1-q}-1}{1-q},$ (31)

where, in the limit $q\to 1$, Tsallis entropy $S_{q}$ given by (30), reduces
to Gibbs entropy $S_{G}$

$S_{G}=-k_{B}\sum_{i}p(i)\ln p(i).$ (32)

In fact, the Tsallis entropy (30) satisfies quite general, nonadditive
composition rule of the following form

$\displaystyle\mathcal{S}_{\mathcal{T}12}=\mathcal{S}_{\mathcal{T}1}+\mathcal{S}_{\mathcal{T}2}+\frac{\lambda}{k_{B}}\mathcal{S}_{\mathcal{T}1}\mathcal{S}_{\mathcal{T}2},$
(33)

for a composite system ”12”, made up of two subsystems ”1” and ”2”. In above
equation, we have defined a new nonextensivity parameter $\lambda=1-q$.

###  III.2 Rényi Entropy

The Rényi entropy Rényi (1959), a measure of entanglement in quantum
information that is additive and preserves event independence, is another
important generalization of the Gibbs-Shannon entropy. It is defined as

$S_{R}=k_{B}\frac{\ln\sum_{i}p^{q}(i)}{1-q}.$ (34)

It is important that $S_{R}$ can be written in terms of $S_{\mathcal{T}}$ by
using the formal logarithm approach Biró and Ván (2011), and both entropies
are related as follows

$\displaystyle
S_{R}=\frac{k_{B}}{\lambda}\ln[1+\frac{\lambda}{k_{B}}\mathcal{S}_{\mathcal{T}}].$
(35)

It is interesting to mention here that $S_{R}$ is the equilibrium entropy
which corresponds to an equilibrium temperature $T_{R}$ defined from the
equilibrium condition by maximizing the Tsallis entropy (33), which is given
by Çimdiker  _et al._ (2022)

$T_{R}=(1+\frac{\lambda}{k_{B}}S_{\mathcal{T}})\frac{1}{k_{B}\beta}.$ (36)

Here, $k_{B}\beta=\partial S_{\mathcal{T}}/\partial U$, where $U$ is the
internal energy of the nonextensive system.

####  III.2.1 Rényi black hole Entropy and Temperature

For the case of a Schwarzschild black hole, assuming that the Bekenstein
entropy $S_{B}$ is just the Tsallis entropy $S_{\mathcal{T}}$, and replacing
internal energy $U$ with the mass of the black hole $M$ in equations (35) and
(36), the Rényi entropy can be defined on the horizon of a black hole as Biró
and Czinner (2013); Czinner (2015); Czinner and Iguchi (2016, 2017b, 2017a)

$S_{R}=\frac{k_{B}}{\lambda}\ln[1+\frac{\lambda}{k_{B}}S_{B}],$ (37)

and the associated Rényi temperature reads

$T_{R}=(1+\frac{\lambda}{k_{B}}S_{B})T_{H}.$ (38)

Furthermore, we can write down the GUP corrected Rényi entropy using GUP
corrected Bekenstein entropy as follows Alonso-Serrano  _et al._ (2021) (cf.
Fig. 5)

$\displaystyle
S_{Rgup}=\frac{k_{B}}{\lambda}\ln\left[1+\frac{\lambda}{k_{B}}(S_{GUP})\right],$
(39)

and corresponding GUP modified Rényi temperature $T_{Rgup}$ can be written as
(cf. Fig. 6)

$\displaystyle
T_{Rgup}=\left[1+\frac{\lambda}{k_{B}}(S_{GUP})\right]\mathcal{K}T_{H}.$ (40)

Figure 5: Rényi entropy $S_{R}$ of a black hole vs its mass $M$. Dashed lines
represent GUP corrected cases, $\lambda\rightarrow 0$ limit is the Bekenstein-
Hawking case.

The Rényi entropy increases logarithmically (for $0<\lambda<1$), whereas the
Bekenstein entropy ($\lambda\rightarrow 0$) increases quadratically, as shown
in Fig. 5. Furthermore, for the GUP corrections, the Rényi black holes do not
completely evaporate; rather, evaporation stops at the critical mass $M_{r}$,
leaving a remnant with finite entropy and temperature as the Rényi black
hole’s final state.

Figure 6: Rényi temperature $T_{R}$ of a black hole vs its mass $M$. Dashed
lines represent GUP corrected cases, $\lambda\rightarrow 0$ limit is the
Bekenstein-Hawking case.

Using (38) and (40), we can write the inverse Rényi temperature parameters,
$\beta_{R}$ and $\beta_{Rgup}$, which will further be used in calculating the
heat capacities, such that

$\displaystyle
k_{B}\beta_{R}=\frac{S^{\prime}_{B}(M)/c^{2}}{1+\frac{\lambda}{k_{B}}S_{B}}=\frac{k_{B}\beta}{1+\frac{\lambda}{k_{B}}S_{B}},$
(41)

and the GUP-corrected inverse Rényi temperature reads

$\displaystyle
k_{B}\beta_{Rgup}=\frac{S^{\prime}_{GUP}(M)/c^{2}}{1+\frac{\lambda}{k_{B}}S_{GUP}}=\frac{k_{B}\beta_{GUP}}{1+\frac{\lambda}{k_{B}}S_{GUP}}.$
(42)

One may determine the characteristic length scale $\mathcal{L_{R}}$ for
$\lambda$ Promsiri  _et al._ (2021, 2020); Nakarachinda  _et al._ (2021),
which reveals the impact of nonextensive parameter $\lambda$ in $S_{R}$ and
$S_{Rgup}$, and in $T_{R}$ and $T_{Rgup}$. As a result, it can be concluded
that below this characteristic length scale $\mathcal{L_{R}}$, the Rényi
temperature behaves like $T_{H}$, and that above $\mathcal{L_{R}}$, the
nonextensive effects increase and $T_{R}$ grows linearly with $M$. The precise
value for the length scale is found in the following subsection.

####  III.2.2 Heat Capacity for the Rényi black hole

In order to investigate the thermodynamic stability of Rényi black holes, we
define the heat capacity $C_{R}$ of the Rényi black hole as

$\displaystyle
C_{R}=-\frac{S^{\prime}{}^{2}_{R}(M)}{S^{\prime\prime}_{R}(M)}.$ (43)

Inserting (41) and (42) into (43), the heat capacity for the non-GUP case
reads

$C_{R}=\frac{C_{Sc}}{1+\frac{\lambda}{k_{B}}S_{B}+\frac{\lambda}{k_{B}}C_{Sc}},$
(44)

and for the GUP case, we have

$\displaystyle
C_{Rgup}=\frac{C_{GUP}}{1+\frac{\lambda}{k_{B}}S_{GUP}+\frac{\lambda}{k_{B}}C_{GUP}}.$
(45)

Figure 7: Heat capacity $C_{R}$ of a Rényi black hole vs its mass $M$. Dashed
lines represent GUP corrected cases, $\lambda\rightarrow 0$ limit is the
Bekenstein-Hawking case.

We plot the heat capacity in Fig. (7), where we can see that $\mathcal{L}$
differentiates two regions for non-GUP and GUP cases. In order to understand
the behavior of $C_{R}$ in both regions, we find $\mathcal{L_{R}}$ in terms of
$\lambda$ from the singular points of equation (44) for the case Schwarzschild
black hole. We find, for the non-GUP case

$\lambda=-\frac{k_{B}}{[S_{B}+C_{Sc}]}=\frac{m_{p}^{2}}{4\pi M^{2}},$ (46)

and for the GUP case, we have

$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle-\frac{k_{B}}{[S_{GUP}+C_{GUP}]}$ $\displaystyle\approx$
$\displaystyle\frac{m_{p}^{2}}{4\pi M^{2}}+\frac{3\alpha m_{p}^{4}}{64\pi
M^{4}}+\frac{\alpha m_{p}^{4}\log\left(\frac{4M}{m_{p}}\right)}{32\pi M^{4}}$

by ignoring the higher order terms in $\alpha$. This means that for the non-
GUP case, we define the mass scale

$M_{c}=\frac{m_{p}}{2\sqrt{\pi\lambda}},$ (48)

which differentiates the two regions and can be further used to define the
characteristic length scale $\mathcal{L}_{R}$, which can be written as

$\mathcal{L}_{R}=2l_{p}\sqrt{\pi\lambda},$ (49)

where we have defined $\mathcal{L}_{R}=GM_{c}/c^{2}$. For the GUP case, we
would expect the characteristic length scale $L_{Rgup}\approx LR+\alpha
f(\lambda)$ by using equation (III.2.2), where $f$ is a function of the
nonextensivity parameter $\lambda$. However, we can not solve it exactly, and
it again shows the effects of $\alpha$ and $\lambda$ for the values of $M$
greater than the GUP corrected mass scale. Interestingly, for the non-GUP
case, the heat capacity is positive for the values greater than this scale,
and below this scale, black holes have negative heat capacity. This means that
black holes with higher masses than $M_{c}$ are thermodynamically stable and
with masses lower than $M_{c}$, they are unstable. Note that, if we exclude
quantum gravity effects, $\mathcal{L}_{R}$ should be greater than $l_{p}$.
This puts a numerical constraint on the nonextensive parameter
$\lambda>1/4\pi$ and this can also be derived by considering $M_{c}>m_{p}$ by
excluding the quantum gravity effects. In Promsiri  _et al._ (2021, 2020);
Nakarachinda  _et al._ (2021), the authors derived this constraint as
$\lambda>1/\pi$ because they considered $\mathcal{L}_{R}=2GM_{c}/c^{2}$ as
characteristic length scale for $\lambda$, where the extra $2$ in
$\mathcal{L}_{R}$ is motivated by Schwarzschild radius $r_{h}=2GM/c^{2}$. We
believe that the proper way to introduce the length or mass scale for
$\lambda$ should be irrespective of the definition which is motivated by
$r_{h}$.

####  III.2.3 Sparsity of the Rényi Radiation

In order to calculate the sparsity profile of Rényi radiation, we replace $T$
with $T_{R}$ in (23), and so the sparsity profile $\eta_{R}$ reads

$\displaystyle\eta_{R}=\frac{\eta_{H}}{[1+\frac{\lambda}{k_{B}}S_{B}]^{2}}.$
(50)

Replacing $T$ with $T_{Rgup}$ and using GUP modified area $A_{GUP}$ in
equation (23), the GUP modified sparsity profile $\eta_{Rgup}$ reads

$\displaystyle\eta_{Rgup}=\frac{\eta_{GUP}}{[1+\frac{\lambda}{k_{B}}S_{GUP}]^{2}}.$
(51)

Figure 8: Sparsity profile $\eta_{R}$ of a Rényi blackhole vs its mass $M$.
Dashed lines represent GUP corrected cases, $\lambda\rightarrow 0$ limit is
the Bekenstein-Hawking case.

From (50), we conclude that the sparsity profile $\eta_{R}$ depends on both
the mass of the black hole and the nonextensivity parameter $\lambda$. From
Fig. (8), we can easily see that the radiation is not sparse initially and
then, at the final stages of the evaporation, the sparsity grows, reaching the
value of $\eta_{H}$, when $M$ approaches to zero. For the GUP case, initially,
the behavior of sparsity is similar to the non-GUP case, however, when $M$
approaches $M_{r}$, it has a finite value which is much less than the sparsity
of Hawking radiation for the non-GUP and GUP cases. Again, we can see the bump
before M reaches $M_{r}$, which is due to the effect of GUP corrections to the
Rényi temperature and GUP corrections to the area.

###  III.3 Tsallis-Cirto Black Hole Entropy

Tsallis-Cirto black hole entropy Tsallis and Cirto (2013) is based on key
principles of Gibbs thermodynamics. First, the entropy must be extensive and
additive, and second, the entropy and associated temperature for a
thermodynamic system must satisfy the Legendre structure. As it was already
said about the Bekenstein entropy in the Introduction, it violates a key
principle of classical Gibbs thermodynamics and so new definitions of entropy
and temperature for black holes are required in order to comply with the
fundamental principles of thermodynamics in the case of $(3+1)$-dimensional
black holes. Therefore, Tsallis and Cirto proposed the following entropy
definition Tsallis and Cirto (2013); Tsallis (2019).

$\displaystyle\frac{S_{\delta}}{k_{B}}=\left(\frac{S_{B}}{k_{B}}\right)^{\delta},$
(52)

where $\delta>0$ is a real parameter and it follows the composition rule for a
composite thermodynamic system, which is given by

$S_{\delta 12}=k_{B}\left[\left(\frac{S_{\delta
1}}{k_{B}}\right)^{1/\delta}+\left(\frac{S_{\delta
2}}{k_{B}}\right)^{1/\delta}\right]^{\delta}.$ (53)

In this context, the $S_{B}$ is additive, and $S_{\delta}$ is nonadditive. For
$\delta=3/2$, $S_{\delta}$ is proportional to the volume for the case of the
Schwarzschild black hole, and so it is an extensive quantity. The
corresponding Tsallis-Cirto temperature can be written by using the Clausius
relation Çimdiker  _et al._ (2022)

$T_{\delta}=\frac{T_{H}}{\delta}\left(\frac{S_{B}}{k_{B}}\right)^{1-\delta},$
(54)

and it scales with $1/M^{2}$ for $\delta=3/2$, i.e., $T_{\delta}\propto
1/M^{2}$, for the case of Schwarzschild black hole. GUP corrections to the
Tsallis-Cirto black hole entropy can be obtained by the GUP corrected
Bekenstein entropy $S_{GUP}$ given by (15) into (52), which results in

$\displaystyle\frac{S_{\delta
gup}}{k_{B}}=\left(\frac{S_{GUP}}{k_{B}}\right)^{\delta},$ (55)

and the corresponding GUP-modified Tsallis-Cirto temperature can be derived
from the Clausius relation, giving

$T_{\delta
gup}=\frac{T_{GUP}}{\delta}\left(\frac{S_{GUP}}{k_{B}}\right)^{1-\delta}.$
(56)

Figure 9: Tsallis-Cirto entropy $S_{T}$ of a black hole vs its mass $M$.
Dashed lines represent GUP-corrected cases in this figure

From the Figs. (9) and (10), it shows that the evaporation process stops at
the critical value $M_{r}$ for the Tsallis-Cirto case when GUP corrections are
included. This means that the final state of the black hole for the Tsallis-
Cirto case is also a remnant with finite entropy and temperature. Generally,
for the non-GUP case, the parameter $\delta$ plays a significant role. For
$\delta>1/2$, the Tsallis-Cirto entropy behaves similarly to Bekenstein
entropy and increases as a power law of mass, whereas for $\delta<1/2$, it
increases with mass sub-linearly. For $\delta=1/2$, the entropy depends
linearly on mass, and in this case, Tsallis-Cirto temperature becomes
constant. Furthermore, the behavior of the Tsallis temperature is similar to
the Hawking temperature for $\delta>1/2$ while for $\delta<1/2$, the behavior
is completely different for the non-GUP case and, interestingly, it behaves
like Rényi temperature for the GUP-corrected case.

Figure 10: Temperature $T_{\delta}$ vs the mass $M$ for Tsallis-Cirto black
hole entropy. Dashed lines correspond to a GUP case.

Note that, unlike $\lambda$ parameter of the Rényi entropy, $\delta$ is not
associated with the length scale for the non-GUP case. On the other hand,
introducing GUP corrections to Tsallis-Cirto entropy, one can define a
characteristic length scale for $\delta$ as well.

####  III.3.1 Heat Capacity for Tsallis-Cirto black holes

Following the previous subsection, the heat capacity for the Tsallis-Cirto
case can be written in terms of $C_{sc}$, and $S_{B}$

$C_{\delta}=C_{Sc}\left[\frac{S_{B}}{S_{B}-(\delta-1)C_{Sc}}\right],$ (57)

where for the Schwarzschild black hole, we have $C_{Sc}=-2S_{B}$. For
$\delta=1/2$, we have infinite heat capacity for all masses. For $\delta<1/2$,
we have positive heat capacity values and negative heat capacity for
$\delta>1/2$. This means that black holes are thermodynamically stable for
$\delta<1/2$, and unstable for $\delta>1/2$.

Figure 11: Heat Capacity $C_{\delta}$ for Tsallis-Cirto black hole entropy.
Dashed lines correspond to a GUP case.

For the GUP corrections, we can write the GUP-corrected heat capacity as

$\displaystyle C_{\delta gup}$ $\displaystyle=$ $\displaystyle
C_{GUP}\left[\frac{S_{GUP}}{S_{GUP}-(\delta-1)C_{GUP}}\right].$ (58)

Note that from equations (15) and (22), we have $-2S_{GUP}\neq C_{GUP}$,
therefore, we can find an associated characteristic length scale
$\mathcal{L}_{\delta gup}$ for the $\delta$ parameter, for which, we have two
regions, which corresponds to positive and negative values of GUP corrected
heat capacities. The length scale $\mathcal{L}_{\delta gup}$ can be found by
using the singular points of the above equation (58) for $\delta$, which is
given by

$\displaystyle\delta=\frac{S_{GUP}}{C_{GUP}}+1.$ (59)

One could solve the above equation (59) for mass $M$, which gives
$\mathcal{L}_{\delta gup}$ as a function of $\delta$. However, it is
analytically not possible. One may use the perturbative approach to solve the
equation for $M$ and define the corresponding length scale or mass scale. From
the Figs. (9) and (11), for $\delta<1/2$, and below $\mathcal{L}_{\delta
gup}$, the GUP corrected Tsallis-Cirto entropy behaves like $S_{R}$ and it
gives positive GUP modified heat capacity for the GUP case. For values
$\delta>1/2$, $\mathcal{L}_{\delta gup}$ does not exist as (59) yields
imaginary numbers. Thus, it gives negative heat capacity, implying that GUP-
corrected Tsallis black holes are thermodynamically stable for $\delta<1/2$,
and unstable for $\delta>1/2$.

####  III.3.2 Sparsity of the Tsallis-Cirto Radiation

By following the previous subsection, and using the Tsallis-Cirto temperature,
we can write the sparsity profile $\eta_{\delta}$ for Tsallis-Cirto radiation
as

$\displaystyle\eta_{\delta}=\eta_{H}\delta^{2}\left(\frac{S_{B}}{k_{B}}\right)^{2\delta-2},$
(60)

and the GUP-corrected sparsity profile $\eta_{\delta gup}$, by using (23) and
(56), it can be written as

$\displaystyle\eta_{\delta
gup}=\eta_{GUP}\delta^{2}\left(\frac{S_{GUP}}{k_{B}}\right)^{2\delta-2}.$ (61)

Figure 12: Sparsity profile $\eta_{\delta}$ for Tsallis-Cirto black hole
entropy. Dashed lines correspond to a GUP case.

Fig. (12) depicts the sparsity profile vs. mass relationship. For the Tsallis-
Cirto temperature, the sparsity scales with $M^{4\delta-4}$. Again, the value
of $\delta$, significantly changes the behavior of the sparsity. It should be
noted that the sparsity parameter is now affected by mass as well as $\delta$
and the GUP-parameter $\alpha$. In the non-GUP case, $\eta_{\delta}=\eta_{H}$
for $\delta=1$. When $\delta>1$, the value of $\eta_{\delta}$ is initially
very high and approaches zero at the end of the black hole evaporation. This
means that, initially, the Tsallis-Cirto radiation is highly sparse, and
during the final stages of evaporation, it is not sparse at all. In this way,
for $\delta<1$, Tsallis-Cirto radiation is initially not sparse, but at the
end of the evaporation, it is extremely sparse with the sparsity parameter
infinite. For the GUP case, initially, the behavior is the same as for the
non-GUP case, but when the mass approaches the order of Planck mass, i.e., the
remnant mass $M_{r}$, the sparsity parameter decreases to some finite values
for each case. Note that all these finite values of sparsity profiles are less
than the standard sparsity profile $\eta_{H}$.

###  III.4 Sharma-Mittal Entropy

Sharma-Mittal (SM) is an entropic form Sharma and Mittal (1977); Masi (2005)
that generalizes the Rényi and Tsallis entropies. It is defined as

$\displaystyle
S_{SM}=\frac{1}{R}\left[\left(\sum_{i=1}^{W}p_{i}^{1-\lambda}\right)^{\frac{R}{\lambda}}-1\right]$
(62)

where $R$ is another free parameter that is introduced in SM entropy. Under
the equiprobability condition of the states Sayahian Jahromi  _et al._ (2018),
the above equation (62) reduces to

$\displaystyle
S_{SM}=\frac{k_{B}}{R}\left[(1+\frac{\lambda}{k_{B}}S_{T})^{R/\lambda}-1\right],$
(63)

where $R\rightarrow\lambda$ limit yields the Tsallis entropy, and
$R\rightarrow 0$ yields Rényi entropy. The Sharma-Mittal entropy obeys the
same general nonextensive composition rule (33). Assuming that the Bekenstein
entropy $S_{B}$ is the same as the Tsallis entropy $S_{\mathcal{T}}$, we can
write $S_{SM}$ for the case of a Schwarzschild black hole as

$\displaystyle
S_{SM}=\frac{k_{B}}{R}\left[(1+\frac{\lambda}{k_{B}}S_{B})^{R/\lambda}-1\right],$
(64)

and replacing $S_{GUP}$ with $S_{\mathcal{T}}$ in equation (63), the GUP
corrected SM entropy $S_{SMgup}$ reads as

$\displaystyle
S_{SMgup}=\frac{k_{B}}{R}\left[(1+\frac{\lambda}{k_{B}}S_{GUP})^{R/\lambda}-1\right].$
(65)

The corresponding temperatures can be found by using the Clausius relation, as

$\displaystyle
T_{SM}=T_{H}(1+\frac{\lambda}{k_{B}}S_{B})^{1-\frac{R}{\lambda}},$ (66)

and the GUP corrected SM temperature $T_{SMgup}$ reads as

$\displaystyle
T_{SMgup}=T_{GUP}(1+\frac{\lambda}{k_{B}}S_{GUP})^{1-\frac{R}{\lambda}}.$ (67)

We can now define the inverse temperature parameters for GUP and non-GUP cases
by using the above equations (66) and (67), which are given, for the non-GUP
case, as

$\displaystyle\beta_{SM}=\frac{S^{\prime}_{SM}}{k_{B}c^{2}}=\beta(1+\frac{\lambda}{k_{B}}S_{B})^{\frac{R}{\lambda}-1},$
(68)

and for the GUP case, as

$\displaystyle\beta_{SMgup}=\frac{S^{\prime}_{SMgup}}{k_{B}c^{2}}=\beta_{GUP}(1+\frac{\lambda}{k_{B}}S_{GUP})^{\frac{R}{\lambda}-1}.\;\;$
(69)

Figure 13: Plot of the Sharma-Mittal entropy for $\lambda=0.7$. Dashed lines
correspond to a GUP case.

Since SM entropy is the generalization of the Tsallis and Rényi entropy, the
behavior of the temperature and the entropy are similar to that of $S_{B}$ and
$S_{R}$ and $T_{H}$ and $T_{R}$ for different values of Sharma-Mittal
parameter $R$. Also, the black hole does not evaporate in this case as well,
and the evaporation process stops at $M_{r}$, leaving the final state of the
black hole as a remnant having finite entropy and temperature. The plots of SM
entropy and temperature are given in Figs. 13 and 14.

Figure 14: Sharma-Mittal temperature for $\lambda=0.7$. Dashed lines
correspond to a GUP case.

####  III.4.1 Heat Capacity for Sharma-Mittal Black Holes

By following the previous subsections, we can calculate the heat capacity
$C_{SM}$ for the SM black holes as

$\displaystyle
C_{SM}=\frac{C_{Sc}(1+\frac{\lambda}{k_{B}}S_{B})^{\frac{R}{\lambda}}}{(1+\frac{\lambda}{k_{B}}S_{B})-\frac{\lambda}{k_{B}}C_{Sc}\left(\frac{R}{\lambda}-1\right)}\;,$
(70)

and for the GUP SM black holes case, it reads as

$\displaystyle
C_{SMgup}=\frac{C_{GUP}(1+\frac{\lambda}{k_{B}}S_{GUP})^{\frac{R}{\lambda}}}{(1+\frac{\lambda}{k_{B}}S_{GUP})-\frac{\lambda}{k_{B}}C_{GUP}\left(\frac{R}{\lambda}-1\right)}\;.$
(71)

The plots of (70) and (71) are given in Fig. 15. Similarly as for the Rényi
case, we define the characteristic length scale $\mathcal{L}_{SM}$ in terms of
$\lambda$ and $R$ by employing the singular point of $C_{SM}$. For the non-GUP
case, we have such a singular point for

$\displaystyle\lambda=\frac{RC_{Sc}-k_{B}}{C_{Sc}+S_{B}}\;.$ (72)

From (72), we can easily define the following characteristic relation by
solving it for $M$, which reads

$\mathcal{L}_{SM}=2l_{p}\sqrt{\pi(\lambda-2R)},$ (73)

where $\mathcal{L}_{SM}=GM_{c}/c^{2}$, and the mass scale $M_{c}$ is defined
as

$M_{c}=\frac{m_{p}}{2\sqrt{\pi(\lambda-2R)}}.$ (74)

Similarly, one can define $\mathcal{L}_{SMgup}$ for the GUP case by using the
following singular point at

$\displaystyle\lambda=\frac{RC_{GUP}-k_{B}}{C_{GUP}+S_{GUP}}\;\;,$ (75)

and solve it for $M$. Since the analytic solution is not possible, one could
use a perturbative approach to find the GUP corrections to $\mathcal{L}_{SM}$
up to the first order in $\alpha$. Note that $R\rightarrow 0$ limit yields the
$\mathcal{L}_{R}$ for the Rényi case. For $\lambda-2R>0$ and $M>M_{c}$, the
heat capacity is positive for both non-GUP and GUP cases, and for $M<M_{c}$,
the heat capacity is negative for both non-GUP and GUP cases.

Figure 15: Heat capacity $C_{SM}$ for Sharma-Mittal entropy for $\lambda=0.7$.
Dashed lines correspond to a GUP case.

####  III.4.2 Sparsity of the Sharma-Mittal Radiation

The sparsity profile $\eta_{SM}$ can be derived by applying the Sharma-Mittal
temperature to (23), and reads

$\displaystyle\eta_{SM}=\eta_{H}(1+\frac{\lambda}{k_{B}}S_{B})^{2(\frac{R}{\lambda}-1)},$
(76)

and for the GUP case, substituting equations (67) and (28) in (23), the GUP
modified sparsity profile for the Sharma-Mittal radiation reads as

$\displaystyle\eta_{SMgup}=\eta_{GUP}(1+\frac{\lambda}{k_{B}}S_{GUP})^{2(\frac{R}{\lambda}-1)}.$
(77)

The plots of the sparsity profile for SM (76) and SM GUP (77) cases are given
in Fig. 16.

Figure 16: Sparsity profile for Sharma-Mittal entropy for $\lambda=0.4$.
Dashed lines correspond to a GUP case.

The behavior of the sparsity profile again depends on the Sharma-Mittal
parameter $R$ in addition to the nonextensive parameter $\lambda$ and also the
GUP parameter $\alpha$ in the case of GUP corrections. For the values of
$\lambda$ and $R$, which satisfy the inequality $\lambda>R$, the sparsity of
the Sharma-Mittal radiation behaves like the sparsity of the Rényi radiation
for both non-GUP and GUP cases. This means that, initially, the Sharma-Mittal
radiation is not sparse, and at the end of the evaporation, its value
approaches the value of Hawking’s case, i.e., $\eta_{H}$, for the non-GUP
case. At the $R\rightarrow 0$ limit we obtain the sparsity profile of the
Rènyi entropy. For $R>\lambda$, initially, the Sharma-Mittal sparsity profile
is higher than $\eta_{H}$ and its value exactly approaches $\eta_{H}$ at the
end of the evaporation, while for the case of GUP, it approaches to some
finite value less than $\eta_{H}$. It is interesting to note that, for
$\alpha>0$, the GUP modified sparsity parameter is always less than the
standard Hawking case.

###  III.5 Kaniadakis Entropy

Kaniadakis entropy Kaniadakis (2002); Drepanou  _et al._ (2022) is a type of
nonextensive entropy that results from the Lorentz transformation of special
relativity. It is a single parameter deformation of Gibbs entropy in which the
standard Gibbs entropy is generalized to the relativistic regime with the help
of a new parameter $K$ that is connected to the dimensionless rest energy of
the various parts of a multibody relativistic system. The Kaniadakis entropy
$S_{K}$ is defined as

$\displaystyle S_{K}=k_{B}\log_{K}\Omega$ (78)

where

$\displaystyle\log_{K}(\Omega)=\frac{\Omega^{K}-\Omega^{-K}}{2K}\;.$ (79)

Considering $S_{B}=k_{B}\ln\Omega$, which means that the number of microstates
$\Omega$ for a black hole is proportional to $e^{S_{B}/k_{B}}$, the above
equation (78) can be written in the following form

$\displaystyle S_{K}=\frac{k_{B}}{K}\sinh\left[K\frac{S_{B}}{k_{B}}\right],$
(80)

where we have used equation (79) for the $\sinh x$ function and used the
relation $\Omega=e^{S_{B}/k_{B}}$. Replacing $S_{B}$ with $S_{GUP}$, the GUP
modified Kaniadakis entropy $S_{KGUP}$ reads as

$\displaystyle
S_{KGUP}=\frac{k_{B}}{K}\sinh\left[K\frac{S_{GUP}}{k_{B}}\right].$ (81)

Figure 17: Kaniadakis Entropy $S_{K}$ vs mass $M$. Dashed lines correspond to
a GUP case.

Note that, in the limit $K\rightarrow 0$, $S_{K}$ reduces to Gibbs entropy. In
Fig. (17), one can see the characteristic form of sine hyperbolic $(\sinh)$
function for different small values of $K$ which shows the similar behaviour
like the Bekenstein entropy. As expected, for the GUP case, black holes do not
evaporate completely and the final state of the black hole is a remnant like
for the case of standard GUP modified Bekenstein-Hawking case. Furthermore, as
$K$ increases, the entropy increases sharply. By using the Clausius relation,
the corresponding Kaniadakis black black hole temperature $T_{K}$ reads as

$T_{K}=T_{H}\,\operatorname{sech}\left[K\frac{S_{B}}{k_{B}}\right],$ (82)

and the GUP modified Kaniadakis temperature $T_{KGUP}$ can be written as

$T_{Kgup}=T_{GUP}\,\operatorname{sech}\left[K\frac{S_{GUP}}{k_{B}}\right].$
(83)

By using (82) and (83), one can write the following inverse temperature
parameters $\beta_{K}$ as follows

$k_{B}\beta_{K}=k_{B}\beta\cosh\left[K\frac{S_{B}}{k_{B}}\right],$ (84)

and for the GUP case, $\beta_{KGUP}$ reads

$\displaystyle
k_{B}\beta_{Kgup}=k_{B}\beta_{GUP}\cosh\left[K\frac{S_{GUP}}{k_{B}}\right],$
(85)

which can further be used to find the heat capacities for Kaniadiakis black
holes.

Figure 18: Kaniadakis temprature $T_{K}$ vs mass. Dashed lines correspond to a
GUP case.

Fig. (18) shows that Kaniadakis temperature behaves as Hawking temperature
with a slight change depending on the parameter $K$. For the GUP case, it
stops at some finite value, when $M$ approaches to $M_{r}$ during the final
stages of the black hole evaporation process.

####  III.5.1 Heat capacity for Kaniadakis Black Holes

The heat capacities for Kaniadakis entropy can be calculated by following the
previous subsections. For the non-GUP case, the heat capacity $C_{K}$ for
Kaniadakis black hole reads as

$\displaystyle
C_{K}=C_{Sc}\frac{\cosh^{2}[K\frac{S_{B}}{k_{B}}]}{\cosh[K\frac{S_{B}}{k_{B}}]-C_{Sc}\sinh[K\frac{S_{B}}{k_{B}}]},$
(86)

and for the GUP modified heat capacity, $C_{Kgup}$, it can written as

$\displaystyle
C_{Kgup}=C_{GUP}\frac{\cosh^{2}[K\frac{S_{GUP}}{k_{B}}]}{\cosh[K\frac{S_{GUP}}{k_{B}}]-C_{GUP}\sinh[K\frac{S_{GUP}}{k_{B}}]}.\;\;\;$
(87)

Figure 19: Kaniadakis heat capacity $C_{K}$ vs mass $M$. Dashed lines
correspond to a GUP case. Figure 20: Sparsity profile $\eta_{K}$ for
Kaniadakis radiation vs mass $M$ of Kaniadakis black hole. Dashed lines
correspond to a GUP case.

From Fig. (19), one can easily notice the negative heat capacities for all
values of $K$. This means that Kaniadakis black holes are thermodynamically
unstable for all $M$.

####  III.5.2 Sparsity profile of the Kaniadakis Radiation

The sparsity profile $\eta_{K}$ for the Kaniadakis radiation can be derived by
applying (82) into (23), and reads

$\displaystyle\eta_{K}=\eta_{H}\cosh^{2}\left(K\frac{S_{B}}{k_{B}}\right),$
(88)

and for the GUP modified sparsity profile $\eta_{KGUP}$, we apply (83) and
(28) into (23), to obtain

$\displaystyle\eta_{KGUP}=\eta_{GUP}\cosh^{2}\left(K\frac{S_{GUP}}{k_{B}}\right).$
(89)

From Fig. (20), the sparsity parameter for the Kaniadakis case is always high
from the beginning of the evaporation process as compared to the standard
Bekenstein Hawking case. However, for the non-GUP case, $\eta_{K}$ approaches
to the value of $\eta_{H}$ at the end of the evaporation. For the GUP case,
again, it approaches to some finite value of sparsity when $M$ approaches
$M_{r}$, which is always less than the sparsity profile $\eta_{H}$.
Furthermore, we see that increasing value of $K$ directly results in sparser
Kaniadakis radiation.

###  III.6 Barrow entropy

Barrow entropy Barrow (2020) is an entropic form that has no statistical
roots, but is closely tied to black hole horizon geometry. It is proposed to
replace the smooth black hole horizon with a fractal of spheres known as a
sphereflake. This structure is distinguished by its fractal dimension $d_{f}$,
where $3\geq d_{f}\geq 2$, and results in an effective horizon area of
$r_{+}{}^{d_{f}}$, where $r_{+}$ is the horizon radius. As a result, in this
scenario, the horizon area is modified, yielding Barrow entropy as below
$S_{Barrow}$

$\displaystyle
S_{Barrow}=k_{B}\left(\frac{A}{A_{p}}\right)^{1+\frac{\Delta}{2}}$ (90)

where $A$ is the horizon area, $A_{p}$ is the Planck area, and $\Delta$ is the
parameter directly tied to the fractal dimension $d_{f}$ through
$\Delta=d_{f}-2$. In this form, $\Delta$ can take values between 0 and 1, and
$\Delta\rightarrow 1$ limit yields maximally fractal structure, where the
horizon area effectively behaves like a $3-$dimensional volume, while
$\Delta\rightarrow 0$ limit yields the well-known Bekenstein area law where no
fractalization occurs. Although Barrow entropy offers a different picture in
the geometrical sense, in its essence, it has the same form as Tsallis-Cirto
entropy. We can see that they are equivalent by making the following
parametrization in Tsallis-Cirto entropy Abreu and Neto (2020)

$\displaystyle\delta\rightarrow 1+\frac{\Delta}{2}$ (91)

Thus, qualitatively, both entropic forms yield the same temperatures and heat
capacities as a function of black hole mass. Similarly, the Tsallis-Cirto
entropy limit $\Delta=1$ ($\delta=3/2$ for $S_{\delta}$) yields an extensive,
but still nonadditive entropy for black holes.

##  IV Summary and discussion

We have investigated the nonextensive thermodynamics of black holes, the
impact of the generalized uncertainty principle on nonextensive thermodynamics
quantities, and the sparsity and GUP-modified sparsity of the radiation in the
nonextensive scenario. We have found that all nonextensive black hole
entropies and associated temperatures have finite values at the end of the
black hole evaporation process due to GUP modifications, indicating the
existence of a remnant at the end of the evaporation. This means that black
holes do not evaporate fully in the nonextensive setup as well. We have also
investigated the sparsity profiles in each nonextensive configuration. Despite
the fact that the behavior of the sparsity parameter varies for each
nonextensive scenario, GUP consistently lowers the radiation sparsity in all
circumstances toward the end of the evaporation process. Even though multiple
nonextensive scenarios have the same temperatures and entropic profiles, we
have demonstrated that the sparsity parameter can be used to distinguish
between them.

We have introduced GUP and GUP-corrected thermodynamic parameters and have
revised otherwise well-known GUP corrected quantities to a better form in
which the two crucial limits - the extensivity limit for $\lambda\to 0$ and
the HUP limit for $\alpha\to 0$ \- are easily identified. Even though GUP
corrections on Rényi entropy in black hole thermodynamics have been researched
in the literature, we presented a full discussion of it in order to help
readers distinguish between various sorts of nonextensive scenarios.
Additionally, we have provided non-perturbative results for each quantity,
with a focus on the Rényi sparsity parameter, which rises (as shown by the
"bump" in Fig. (8)) before the value of the remnant mass. This is because it
is assumed that the area can change as a result of the GUP-modified Bekenstein
entropy, which is explicitly shown in (29). This indicates that $A_{GUP}$ as
well as $T_{GUP}$ have an impact on the sparsity parameter. Furthermore, we
have introduced black hole mass scale $M_{c}=m_{p}/2\sqrt{\pi\lambda}$ for the
nonextensive parameter $\lambda$ for the Rényi black hole quantities and we
defined corresponding characteristic length for $\lambda$ in terms of $M_{c}$,
i.e. $\mathcal{L}_{R}=GM_{c}/c^{2}=2l_{p}\sqrt{\pi\lambda}$. We have shown
that, for $M>M_{c}$, the heat capacity is positive and hence black holes in
Rényi scenario are thermodynamically stable, while for $M<M_{c}$, the heat
capacity is negative and $S_{R}$ and $T_{R}$ behave like Bekenstein entropy
$S_{B}$ and Hawking temperature $T_{H}$, hence unstable black holes.

Similarly, we have also analyzed the thermodynamic black hole quantities
associated with Tsallis-Cirto black hole entropy. Particularly, we have
focused on GUP corrections and the sparsity of the Tsallis-Cirto radiation. We
have shown that, when GUP corrections are included, Tsallis-Cirto entropy and
associated temperature have a finite value, and this proves that the final
state of the black hole is also a remnant with finite entropy and temperature.
It is interesting to note that the Tsallis-Cirto parameter $\delta$ plays a
significant role. We have found that, for $\delta>1/2$, Tsallis-Cirto entropy
and temperature behave similarly to Bekenstein entropy and Hawking
temperature, and hence have negative heat capacity. For the GUP case, Tsallis-
Cirto temperature behaves like Rényi temperature and has positive heat
capacity for $\delta<1/2$. This means that, in this framework, we must have
$\delta<1/2$ for thermodynamic stability of black holes. In this way, we have
shown that the Tsallis-Cirto sparsity parameter is very high during the start
of the evaporation for $\delta>1$, but it approaches zero at the the end of
the black hole evaporation. On the contrary, for $\delta<1$, we have shown
that the Tsallis-Cirto radiation is not sparse during the start of the
evaporation, but at the end of the evaporation, the sparsity parameter becomes
infinite and hence shows the highly sparse Tsallis-Cirto radiation. The
behavior of the GUP case is initially the same as that of the non-GUP case,
but as the mass approaches the order of Planck mass, i.e., $M_{r}$, the
Tsallis-Cirto sparsity parameter for each case reduces to some finite values.
It should be noted that all of these finite sparsity parameter values are less
than the sparsity parameter $\eta_{H}$ for the standard Hawking case.

We have also shown that the behavior of the temperature and the entropy for
the Sharma-Mittal case is comparable to that of $S_{B}$ and $S_{R}$ and
$T_{H}$ and $T_{R}$ for different values of the Sharma-Mittal parameter $R$
since the Sharma-Mittal entropy is the extension of the Tsallis and Rényi
entropy. Also, in this instance, the black hole does not evaporate, and the
evaporation process stops at $M_{r}$, leaving the black hole in its ultimate
state as a remnant of mass $M_{r}$ with finite entropy and temperature. We
have analysed the sparsity of the Sharma-Mittal radiation and compared it with
the standard Hawking case. We have found that the sparsity of the Sharma-
Mittal radiation behaves similarly to the Rényi radiation in both non-GUP and
GUP instances for values of $\lambda$ and $R$ that fulfill the condition
$\lambda-2R>0$. This indicates that the Sharma-Mittal radiation is initially
not sparse and that by the end of the evaporation, its value approaches that
of Hawking’s scenario, or $\eta_{H}$, for the non-GUP case. When $M$
approaches $M_{r}$ for the GUP case, the Sharma-Mittal sparsity parameter
approaches a finite value that is smaller than $\eta_{H}$. For the case,
$R>\lambda$, we have shown that the Sharma-Mittal sparsity parameter is
initially larger than $\eta_{H}$ and its value exactly approaches $\eta_{H}$
by the end of the evaporation whereas for the case of GUP, it approaches a
finite value that is smaller than $\eta_{H}$. It is noteworthy to notice that,
for $\alpha>0$, the GUP modified sparsity parameter is always lower than the
standard Hawking case. Moreover, we have also introduced the characteristic
mass scale, $M_{c}=m_{p}/2\sqrt{\pi(\lambda-2R})$, for the Sharma-Mittal
scenario and also, defined the corresponding characteristic length scale
$\mathcal{L}_{SM}=GM_{c}/c^{2}=2l_{p}\sqrt{\pi(\lambda-2R)}$. We have shown
that, for $M>M_{c}$ with $\lambda-2R>0$, the black holes are thermodynamically
stable in the Sharma-Mittal scenario for both GUP and non-GUP cases, while for
$M<M_{c}$, black holes are thermodynamically unstable.

We have also examined the Kaniadakis thermodynamic black hole quantities, and
the results demonstrate that, with a little variation depending on the
parameter $K$, Kaniadakis entropy and temperature behave similarly to
Bekenstein entropy and Hawking temperature. In the case of the GUP, both
quantities reach a finite value as black hole mass approaches $M_{r}$ during
the late stages of the black hole evaporation process. It results in negative
heat capacity for all values of $K$, indicating that Kaniadakis black holes
are thermodynamically unstable for all values of black hole mass. Furthermore,
in contrast to the typical Hawking example, the sparsity parameter for the
Kaniadakis instance is consistently high from the beginning of the evaporation
process. For the non-GUP example, however, $\eta_{K}$ approaches the value of
$\eta_{H}$ at the end of the evaporation. In the GUP situation, it approaches
some finite value of sparsity when $M$ approaches $M_{r}$, which is always
smaller than the sparsity parameter $\eta_{H}$. Additionally, it is clear that
a rise in the value of $K$ causes the Kaniadakis radiation to become sparser.

Our short look onto the Barrow entropy has proven its equivalence (though in a
restricted range of parameters) to the Tsallis-Cirto entropy. In view of that,
all the discussion of thermodynamical quantities for Barrow entropy should be
the same as for Tsallis-Cirto.

The main assumption of a nonextensive setup is based on considering Bekenstein
entropy as Tsallis entropy. Therefore in calculations for sparsities, a
Planckian distribution is assumed for all nonextensive entropies. Therefore a
more in-depth study of sparsities of nonextensive statistics can be done by
considering corresponding proper statistics. However, Planck distribution is
the only methodological way to obtain the temperature as surface gravity.
Therefore assuming a different statistical distribution(such as q-distribution
for Tsallis statistics) would again be an educated guess at best. Thus,
current calculations for sparsities are adequate for qualitative
phenomenological assessment.

It is also worth mentioning that we have consequently defined in the paper the
temperatures which were related to the appropriate entropies. It is the fact
that some of the temperatures and entropies are generalising others (like
Sharma-Mittal which generalises Tsallis and Rényi. An open issue remains as
which of these temperatures and entropies have firm physical relevance. This
problem will be addressed elsewhere.

Finally, it is important to emphasize that our conclusions mainly apply to the
simplest spherically symmetric Schwarzschild black holes and may not be
appropriate for physically more advanced (rotating, hairy, multidimensional
etc.) objects automatically.

## Acknowledgments

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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