Barrow Entropy in Fractal Horizons

• Barrow entropy is a generalized horizon entropy that incorporates fractal deformations due to quantum-gravitational effects, replacing the standard area law with a power-law scaling.
• It alters black-hole thermodynamics by modifying temperature, entropy, and evaporation times, and introduces non-equilibrium features.
• In cosmology, Barrow entropy leads to modified Friedmann equations and dark energy models, linking fractal horizon geometry to cosmic evolution.

Barrow entropy is a generalized horizon entropy in which quantum-gravitational effects deform an otherwise smooth black-hole or cosmological horizon into an intricate, fractal-like surface, thereby replacing the Bekenstein–Hawking area law by a power law in the horizon area. In much of the literature surveyed here, the entropy is written as

$$S_B=\left(\frac{A}{4A_P}\right)^{1+\frac{\Delta}{2}},$$

with $0\le \Delta\le 1$, so that $\Delta=0$ recovers the standard area law and $\Delta=1$ corresponds to maximal deformation; one early equipartition analysis instead uses the notation $S_B=(A/4G)^{1+\Delta}$, reflecting a different normalization convention rather than a different physical motivation (Bolotin et al., 2024, Saridakis et al., 2020, Abreu et al., 2020). Across these formulations, the central idea is the same: entropy scales faster than linearly with area because the effective horizon geometry is rough, fractal, or foam-like.

1. Conceptual basis and formal definitions

In standard black-hole thermodynamics, the Bekenstein–Hawking entropy is

$S_{BH}=\frac{A}{4G},$

or equivalently $S_{BH}=A/(4A_P)$ when the Planck area $A_P\sim l_P^2$ is used. Barrow’s proposal modifies this by assigning the horizon an effective fractal deformation induced by quantum-gravitational fluctuations, so that the effective area scales as $r_g^{2+\Delta}$, with $\Delta$ a deformation parameter in the interval $0\le \Delta\le 1$0 (Bolotin et al., 2024). In this form, $0\le \Delta\le 1$1 gives a smooth two-dimensional surface, while $0\le \Delta\le 1$2 gives a maximally deformed horizon whose effective scaling approaches that of a three-dimensional volume (Bolotin et al., 2024).

The form most frequently used in cosmology and in later black-hole applications is

$0\le \Delta\le 1$3

or equivalently

$0\le \Delta\le 1$4

with $0\le \Delta\le 1$5 identified with the Planck area. In these conventions, Barrow entropy reduces exactly to Bekenstein–Hawking entropy at $0\le \Delta\le 1$6, while at $0\le \Delta\le 1$7 it scales as $0\le \Delta\le 1$8 (Bolotin et al., 2024, Saridakis et al., 2020). An important feature emphasized repeatedly is that this is not the usual logarithmically corrected entropy of loop-quantum-gravity or conformal-field-theory type; rather, it is a power-law deformation motivated by geometric fractality of the horizon (Saridakis et al., 2020).

Several works note that the resulting entropy has a Tsallis-like or nonextensive form, in the sense that it is a nonlinear power of the area. At the same time, they also stress that the physical origin is different: Barrow entropy is tied to horizon geometry and quantum-gravitational roughness, not to nonextensive statistical mechanics per se (Saridakis et al., 2020, Bolotin et al., 2024). This distinction matters because later generalizations explicitly combine Barrow deformation with Tsallis nonextensivity, yielding hybrid constructions rather than simple reinterpretations (Bolotin et al., 12 Feb 2026).

2. Black-hole thermodynamics

For a Schwarzschild black hole,

$0\le \Delta\le 1$9

so the standard Barrow form used in later thermodynamic analyses becomes

$\Delta=0$0

From this, the temperature is

$\Delta=0$1

and the number of horizon degrees of freedom is taken to be $\Delta=0$2 (Capozziello et al., 22 Jan 2025). In this convention, increasing $\Delta=0$3 makes the temperature fall faster with mass and increases the entropy at fixed $\Delta=0$4, while the Barrow black hole remains thermodynamically unstable because the heat capacity stays negative in the physically motivated range $\Delta=0$5 (Capozziello et al., 22 Jan 2025).

A closely related but notationally different treatment connects Barrow entropy to the equipartition theorem. Using

$\Delta=0$6

the horizon energy is found to satisfy

$\Delta=0$7

instead of the standard $\Delta=0$8. In that analysis, positivity of the heat capacity requires $\Delta=0$9, which does not overlap with the physically motivated Barrow interval $\Delta=1$0; the physically relevant Barrow black holes therefore remain thermodynamically unstable, just as the ordinary Schwarzschild black hole does (Abreu et al., 2020). This suggests that Barrow deformation modifies the microscopic thermodynamic bookkeeping without removing the basic instability of asymptotically flat Schwarzschild thermodynamics.

Later work extends this picture to logarithmic and nonextensive deformations built on top of Barrow entropy. A logarithmic-corrected Barrow entropy adds loop-quantum-gravity-type terms,

$\Delta=1$1

while a Tsallis-log-corrected Barrow entropy replaces the ordinary logarithm by a $\Delta=1$2-logarithm and introduces a nonextensivity parameter $\Delta=1$3 (Capozziello et al., 22 Jan 2025). In the Schwarzschild sector examined there, these generalized Barrow-based entropies change temperature, equipartition, Helmholtz free energy, and evaporation time, but they do not produce positive heat capacity in the representative parameter ranges studied (Capozziello et al., 22 Jan 2025).

Barrow corrections do, however, modify evaporation time scales. In the standard Barrow Schwarzschild treatment, the lifetime scales as

$\Delta=1$4

so larger $\Delta=1$5 leads to longer-lived black holes; combined logarithmic and Barrow corrections extend the lifetime further (Capozziello et al., 22 Jan 2025). This is consistent with the broader interpretation that fractalized horizons carry more effective degrees of freedom and alter the thermodynamic response of the black hole without changing its qualitative endpoint in the Schwarzschild case (Capozziello et al., 22 Jan 2025).

3. Cosmological dynamics from horizon thermodynamics

A major branch of the literature applies Barrow entropy to the apparent horizon of an FLRW universe via the gravity–thermodynamics conjecture. In this framework one uses the first law

$\Delta=1$6

or its equivalent horizon form, with the apparent-horizon radius

$\Delta=1$7

horizon area $\Delta=1$8, and the modified entropy assigned to the horizon rather than to a black-hole event horizon (Luciano et al., 2022, Sheykhi, 2021). For a general entropy $\Delta=1$9, the resulting Friedmann equations are modified through $S_B=(A/4G)^{1+\Delta}$0, and specialization to Barrow entropy yields extra powers of $S_B=(A/4G)^{1+\Delta}$1 and hence a nonstandard expansion history (Luciano et al., 2022).

One cosmological formulation gives

$S_B=(A/4G)^{1+\Delta}$2

and

$S_B=(A/4G)^{1+\Delta}$3

with

$S_B=(A/4G)^{1+\Delta}$4

so that the standard Friedmann equations are recovered as $S_B=(A/4G)^{1+\Delta}$5 (Luciano et al., 2022). Another derivation rewrites the modification as an effective gravitational coupling $S_B=(A/4G)^{1+\Delta}$6 and yields a Barrow-modified acceleration condition $S_B=(A/4G)^{1+\Delta}$7, showing explicitly that a more negative equation of state is required for acceleration as the deformation grows (Sheykhi, 2021).

A distinct non-equilibrium derivation starts from a modified Einstein-like equation,

$S_B=(A/4G)^{1+\Delta}$8

and in a flat FLRW background arrives at the compact background relation

$S_B=(A/4G)^{1+\Delta}$9

In this form, Barrow entropy modifies both background evolution and linear perturbations, and the case $S_{BH}=\frac{A}{4G},$0 reduces to wCDM (Asghari et al., 2021).

Because these cosmological constructions identify the horizon entropy with a modified apparent-horizon entropy, Barrow entropy also enters holographic and agegraphic dark-energy models. In Barrow holographic dark energy,

$S_{BH}=\frac{A}{4G},$1

with $S_{BH}=\frac{A}{4G},$2 taken as the future event horizon, and the full dynamical system for $S_{BH}=\frac{A}{4G},$3 depends explicitly on $S_{BH}=\frac{A}{4G},$4 (Denkiewicz et al., 2023). In agegraphic dark energy inspired by modified Barrow entropy, the energy density scales as $S_{BH}=\frac{A}{4G},$5 or $S_{BH}=\frac{A}{4G},$6 instead of $S_{BH}=\frac{A}{4G},$7 or $S_{BH}=\frac{A}{4G},$8, and the modified Friedmann equations imply transitions between decelerated and accelerated expansion, with the dark-energy equation of state crossing between quintessence and phantom regimes depending on $S_{BH}=\frac{A}{4G},$9 and the interaction parameter $S_{BH}=A/(4A_P)$0 (Sheykhi et al., 2023).

4. Generalized second law, non-equilibrium behavior, and emergent cosmic space

The thermodynamic consistency of Barrow entropy in cosmology is not settled by a single universal result. One analysis of the generalized second law in a flat FRW universe with matter and dark energy writes the horizon entropy as

$S_{BH}=A/(4A_P)$1

so that

$S_{BH}=A/(4A_P)$2

In the standard case $S_{BH}=A/(4A_P)$3, this reduces to

$S_{BH}=A/(4A_P)$4

so the generalized second law is automatically satisfied. For $S_{BH}=A/(4A_P)$5, however, the sign can depend on the cosmic history; the law remains valid for the $S_{BH}=A/(4A_P)$6CDM background used there, but can be violated for power-law expansion with $S_{BH}=A/(4A_P)$7 when $S_{BH}=A/(4A_P)$8 is sufficiently large (Saridakis et al., 2020). This suggests that Barrow deformation can spoil the exact compensation between fluid entropy and horizon entropy that is present in the standard area-law case.

A different treatment finds that the generalized second law remains valid when one includes matter entropy inside the apparent horizon and uses a modified Friedmann system derived from Barrow entropy. In that analysis,

$S_{BH}=A/(4A_P)$9

so the total entropy never decreases (Sheykhi, 2021). Taken together, these results suggest that the generalized-second-law verdict is framework-dependent: it depends on whether Barrow entropy is inserted into standard GR thermodynamics or into a fully modified gravitational dynamics, and on the assumptions made about equilibrium and the horizon temperature.

That issue is made explicit in work on the emergence of cosmic space under non-equilibrium thermodynamic conditions. For an $A_P\sim l_P^2$0-dimensional non-flat universe with apparent horizon, Barrow entropy leads to a modified law of emergence in both equilibrium and non-equilibrium forms. The central conclusion is that, in order to hold the energy-momentum conservation, the universe with Barrow entropy as the horizon entropy should have non-equilibrium behaviour with an additional entropy production; however, the additional entropy production rate decreases over time, so the system eventually approaches equilibrium (P et al., 2023). A plausible implication is that Barrow entropy generically introduces an irreversible thermodynamic sector, even when the late-time evolution approaches an effectively equilibrium regime.

5. Spacetime foam, measurement limits, and information processing

Barrow entropy has also been used as a phenomenological bridge between fractal horizon geometry and spacetime foam. In that setting, the deformation parameter $A_P\sim l_P^2$1 controls the degree of fractality of the horizon and modifies holographic-type bounds on the number of degrees of freedom in a region of size $A_P\sim l_P^2$2. Replacing the standard black-hole entropy by the Barrow form changes

$A_P\sim l_P^2$3

into

$A_P\sim l_P^2$4

which yields the generalized measurement uncertainties

$A_P\sim l_P^2$5

For $A_P\sim l_P^2$6, these reproduce the Karolyhazy/Ng relations $A_P\sim l_P^2$7 and $A_P\sim l_P^2$8; for $A_P\sim l_P^2$9, they collapse to Planck-scale lower bounds independent of the macroscopic interval, $r_g^{2+\Delta}$0 and $r_g^{2+\Delta}$1 (Bolotin et al., 2024).

Within the same framework, the time-resolution bound can be reinterpreted as a bound on information processing. Writing $r_g^{2+\Delta}$2 as the cycle time and $r_g^{2+\Delta}$3 as the maximum number of elementary processing steps, one obtains

$r_g^{2+\Delta}$4

Here larger $r_g^{2+\Delta}$5 tightens the combined constraint on processing speed and total processed information (Bolotin et al., 2024). The same analysis links increased fractality to a shorter available time interval for a black-hole “clock,” so that higher $r_g^{2+\Delta}$6 both increases entropy and reduces the lifetime available for information processing in that clock interpretation (Bolotin et al., 2024). This supports the description of Barrow entropy as a geometric-information bridge between horizon microstructure, spacetime discreteness, and computational bounds.

6. Observational status and phenomenology

Observational analyses of Barrow entropy are highly nonuniform and produce some of the sharpest tensions in the subject. In gravitational baryogenesis driven by Barrow-modified Friedmann equations, matching the observed baryon asymmetry requires

$r_g^{2+\Delta}$7

provided the baryon asymmetry is generated specifically by the Barrow-induced gravitational baryogenesis mechanism (Luciano et al., 2022). Big Bang nucleosynthesis is much more restrictive: requiring that the deviation of the weak freeze-out temperature remain within observational limits gives

$r_g^{2+\Delta}$8

so any constant Barrow deformation must be extremely small not to spoil the BBN epoch (Barrow et al., 2020).

Late-time cosmological tests lead to more varied conclusions. In a model-independent reconstruction using cosmographic parameters, the deformation is related to curvature quantities through

$r_g^{2+\Delta}$9

and for $\Delta$0 this reduces to

$\Delta$1

That analysis predicts $\Delta$2 and argues that very precise measurements of the third derivative of the scale factor could directly test the Barrow scenario (Salehi, 2023). In a full modified-cosmology implementation confronted with CMB, supernovae, BAO, lensing, and RSD data, Barrow cosmology is found to be compatible with wCDM and can slightly reduce either the $\Delta$3 tension or the $\Delta$4 tension depending on the dataset combination and on whether the preferred dark-energy sector is phantom or quintessential, but the best-fit $\Delta$5 remains small and compatible with $\Delta$6 within uncertainties (Asghari et al., 2021).

A very different conclusion is reached in Barrow entropic holographic dark energy. Using the full set of dynamical and geometrical late-time data, that framework points toward a nearly extensive Gibbs-like entropic behaviour with

$\Delta$7

close to the maximal Barrow value $\Delta$8, and excludes the standard Bekenstein area-entropy limit $\Delta$9 within that model (Denkiewicz et al., 2023). This directly contradicts the early-universe limits from baryogenesis, BBN, and inflation emphasized in the same discussion (Denkiewicz et al., 2023). The contrast suggests that current constraints are strongly model-dependent: Barrow entropy used as the source of a holographic dark-energy sector is not constrained in the same way as Barrow entropy used as a small correction to early-universe thermodynamics.

Barrow effects have also been examined in stochastic gravitational waves from a first-order cosmological QCD phase transition. In that application, Barrow entropy modifies the temperature–Hubble relation and the scale-factor–temperature relation, shifting the gravitational-wave signal toward the lower-frequency regime; for an observationally suggested value $0\le \Delta\le 1$00, the predicted stochastic background lies within the sensitivity bands of SKA, IPTA, EPTA, and NANOGrav 12.5-year observations (Feng et al., 2022). This suggests that Barrow entropy may have testable consequences beyond background cosmology, especially in PTA-band gravitational-wave phenomenology.

7. Generalizations and related frameworks

Barrow entropy has been generalized in several directions. One major extension combines it with Tsallis nonextensivity to form the Barrow–Tsallis entropy

$0\le \Delta\le 1$01

with $0\le \Delta\le 1$02 the nonextensivity parameter. In that framework, the corresponding holographic dark-energy density is

$0\le \Delta\le 1$03

and cosmography yields the exact relation

$0\le \Delta\le 1$04

The same paper also considers an extended range $0\le \Delta\le 1$05 in the hybrid setting, motivated by the possibility of voids or porosity in the effective horizon geometry (Bolotin et al., 12 Feb 2026). This shows that once Barrow deformation is embedded in a broader generalized-entropy family, the original interval $0\le \Delta\le 1$06 need not remain the only phenomenological option.

In modified-gravity applications, Barrow entropy has been combined with $0\le \Delta\le 1$07 gravity through the gravity–thermodynamics conjecture. The merged framework leads to Friedmann equations containing both $0\le \Delta\le 1$08 and the Barrow factor $0\le \Delta\le 1$09, and the combined dark-energy sector depends on both curvature and fractal-area corrections (Ens et al., 2022). This suggests that Barrow entropy can be treated either as an alternative to modified gravity or as a thermodynamic sector superposed on it.

Black-hole extensions beyond four-dimensional Schwarzschild space are numerous. For $0\le \Delta\le 1$10-dimensional Gauss–Bonnet black holes, the Barrow–Gauss–Bonnet entropy is

$0\le \Delta\le 1$11

with a correspondingly modified temperature. In the analysis reported, five-dimensional black holes have both stable and unstable branches, while $0\le \Delta\le 1$12 black holes keep negative heat capacity and evaporate completely despite Barrow and Gauss–Bonnet corrections (Shi et al., 26 Aug 2025). For brane-world black holes, Barrow entropy produces divergence points in the heat capacity and modifies thermodynamic topology; fixing deformation and cosmological parameters gives a topological charge $0\le \Delta\le 1$13 predominantly controlled by the dark matter parameter, while in the de Sitter model the cosmological horizon prevents stable photon spheres (Zafar et al., 1 Mar 2026). In an RN–AdS black hole with cloud of strings and quintessence, the Smarr relation becomes

$0\le \Delta\le 1$14

and the non-zero topological charge is taken as indicating the presence of a critical point (Zafar et al., 1 Apr 2025).

A recurring misconception is that all generalized entropies used in gravity are interchangeable. The literature summarized here does not support that view. Barrow entropy is repeatedly described as mathematically similar to Tsallis-type expressions but physically rooted in fractal deformation of horizon geometry rather than in a nonextensive composition rule for probabilities (Saridakis et al., 2020, Bolotin et al., 2024). The persistence of distinct phenomenology under Barrow-only, Barrow–Tsallis, Barrow+$0\le \Delta\le 1$15, and Barrow–Gauss–Bonnet constructions suggests that the geometric interpretation of $0\le \Delta\le 1$16 remains the defining feature even when the entropy is embedded in a broader generalized-thermodynamic setting.