Generalized Horizon Entropy

• Generalized horizon entropy is an extension of conventional black hole entropy that incorporates deformation parameters to include nonadditive statistical effects and quantum corrections.
• It alters gravitational dynamics by modifying the Friedmann equations and introducing effective dark energy, thereby unifying cosmic inflation with late-time acceleration.
• The framework connects to modified gravity theories and entropy quantization, providing new insights into thermodynamics, cosmic evolution, and holographic principles.

Generalized horizon entropy refers to extensions and deformations of the standard Bekenstein–Hawking entropy associated with black hole, cosmological, or apparent horizons. Such generalizations are constructed to remain consistent with the Clausius relation, incorporate nonadditive statistical frameworks, address quantum and statistical gravitational phenomena, and implement the holographic principle in a broader class of gravitational and cosmological theories. These modifications fundamentally impact the thermodynamics of spacetime, the form of the Friedmann equations, and the classification of cosmic acceleration and dark energy phenomena.

1. Formal Definitions and Construction Principles

The Bekenstein–Hawking entropy $S_{BH} = A/(4G)$, with $A$ the horizon area, is recovered in Einstein gravity with local horizon thermodynamics. Generalized horizon entropies arise by altering the functional relation between horizon mass and size or by introducing statistical nonadditivity, leading to expressions such as: $$S = f(S_{BH}) \quad \text{or} \quad S = S_{BH} \cdot g(S_{BH}) \quad \text{or via a generalized mass-to-horizon relation.}$$

A notable example is the nonadditive entropy (Kruglov, 12 Feb 2025): $S_K = \frac{S_{BH}}{1 + \gamma S_{BH}}$ where $\gamma$ is a free parameter (with $[\text{entropy}]^{-1}$ units). In the microcanonical limit, $S_K$ emerges from a Gibbs-like generalized entropy with equiprobability $p_i=1/W$: $$S_K = \frac{\ln W}{1 + \gamma \ln W}, \qquad S_{BH} = \ln W.$$ For $\gamma \to 0$, $S_K \to S_{BH}$. Generalized horizon entropies are typically nonadditive: $S_K(A+B) \ne S_K(A) + S_K(B).$

This form is one alternative to generalizations evaluated in the nonextensive (Tsallis, Rènyi) and fractal (Barrow, Kaniadakis) frameworks, all retaining dependence on $S_{BH}$ or horizon geometric invariants (Saridakis et al., 2020, Abreu et al., 2021, Prasanthan et al., 5 Oct 2024).

2. Gravitational Dynamics and Generalized First Law

In FRW spacetimes, the horizon entropy determines modifications to the cosmological field equations via the first law of horizon thermodynamics: $dE = -T_h dS_h + W dV_h.$ Here, $E=\rho V_h$, $W = (\rho - p)/2$, and $V_h$ is the horizon volume. The horizon temperature is determined kinematically, e.g., via the Kodama–Hayward form (Kruglov, 12 Feb 2025): $$T_h = \frac{H}{2\pi} \left| 1 + \frac{\dot H}{2H^2} \right|.$$ For generalized entropy $S_h(H)$, the time-evolution of Hubble rate is governed by a Raychaudhuri-type equation: $$\frac{\dot H}{[1 + \gamma \pi/(G H^2)]^2} = -4\pi G (\rho + p)$$ which yields a nontrivial first Friedmann equation (with $b \equiv \pi \gamma / G$): $$H^2 - \frac{b^2}{H^2 + b} - 2b\ln\left(\frac{H^2 + b}{b}\right) = \frac{8\pi G}{3}\, \rho$$ There is a strict reduction to standard cosmology as the deformation parameter ($\gamma$) vanishes.

3. Dark Sector Realizations and Effective Cosmological Parameters

The generalized entropy induces an effective "holographic dark energy" component characterized by: $H^2 = \frac{8\pi G}{3} (\rho + \rho_D)$ with

$$\rho_D = \frac{3b}{8\pi G}\left[ \frac{b}{H^2 + b} + 2 \ln \left( \frac{H^2 + b}{b} \right) \right]$$

and effective pressure

$$p_D = - \frac{b(b+2H^2)\dot H}{4\pi G(b+H^2)^2} - \rho_D$$

The equation-of-state for the dark sector,

$w_D = \frac{p_D}{\rho_D},$

asymptotes to $w_D \to -1$ at large $H$, mimicking a cosmological-constant-like phase.

The deceleration parameter, which captures the transition between cosmic deceleration and acceleration, takes the generalized form: $$q \equiv -1 - \frac{\dot H}{H^2} = -1 + \frac{3(1+w)(H^2 + b)^2}{2H^6}\left( H^2 - \frac{b^2}{H^2 + b} - 2b\ln \frac{H^2 + b}{b} \right)$$ The parameter space allows for either permanent acceleration ($w < -1/3$) or a deceleration-to-acceleration transition ($-1/3 < w < 0$), encoding a unified description of cosmic inflation and late-time acceleration within a single entropy-based modification (Kruglov, 12 Feb 2025).

4. Structural Equivalence with Modified Gravity Theories

Kruglov's entropy model is dynamically equivalent to $F(T)$ teleparallel gravity, with $T = -6H^2$ the torsion scalar and $F_T \equiv dF/dT$: $$H^2 = \frac{8\pi G}{3} \rho - \frac{F}{6} + \frac{T F_T}{3}$$ Matching the effective dark energy sector determines $F(T)$: $$F(T) = T + 3b \frac{b}{(-T/6) + b} + 6b\ln \frac{(-T/6) + b}{b}$$ This demonstrates that background FRW cosmology governed by a generalized horizon entropy can be mapped exactly onto a subclass of modified torsion gravity models (Kruglov, 12 Feb 2025).

5. Thermodynamics, the Second Law, and Parameter Constraints

Nonadditive or deformed horizon entropies generically jeopardize the monotonicity of the total entropy production rate. For the specific case $S_K$, the combined system (matter plus horizon) obeys the generalized second law provided that the deformation parameter $\gamma$ is very small. Matching deceleration parameter and transition redshift to cosmological observations typically fixes $b = \pi \gamma / G \sim 0.4 - 0.6$, corresponding to present values $q_0 \sim -0.6$ and transition redshift $z_t \sim 0.6 - 0.8$ (Kruglov, 12 Feb 2025).

Any substantial departure in the deformation parameter ($\gamma$) from zero leads to disagreements with Big Bang nucleosynthesis and CMB constraints. This suggests that only minimal nonadditive generalizations are compatible with precision cosmology.

6. Spectral Properties, Quantization, and Universality

Generalized horizon entropies can be quantized semi-classically. For spherically symmetric horizons in theories where the entropy is a function of Wald entropy, the entropy spectrum is universally evenly spaced (Skakala et al., 2013): $S_n = n\, \epsilon, \quad \epsilon = 2\pi\hbar,$ regardless of the precise functional deformation, provided a first law applies and a Rindler limit exists.

This universality in entropy level spacing has significant implications: the Bekenstein quantization of entropy is robust under a wide class of entropy deformations (e.g., higher-curvature corrections, nonadditive statistics), although the corresponding area spectrum loses its equidistant character if $S \not\propto A$.

7. Implications and Unified Cosmological Evolution

A key phenomenological consequence of generalized horizon entropy is the appearance of a dynamical effective cosmological constant: $$\Lambda_{\rm eff} = \frac{3b^2}{H^2+b} + 6b \ln \frac{H^2 + b}{b}$$ $\Lambda_{\rm eff}(H)$ decreases monotonically with $H$, interpolating between $\Lambda_{\rm eff} \to 3b$ at late times ($H \to 0$) and vanishing at early times ($H \gg b$), allowing for the dynamical realization of inflationary and dark energy epochs within a single theoretical scheme.

The background cosmological dynamics, scalar perturbation evolution, and structure formation metrics such as the growth factor and primordial gravitational wave background are all sensitive functions of the generalized entropy parameters (Luciano, 1 Oct 2025, Luciano et al., 18 Aug 2025).

This suggests that nonadditive entropy models are subject to stringent cosmological and astrophysical constraints, but also offer a unified phenomenological framework for early- and late-time acceleration.

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References:

• "New entropy, thermodynamics of apparent horizon and cosmology" (Kruglov, 12 Feb 2025)
• "Horizon spectroscopy in and beyond general relativity" (Skakala et al., 2013)
• See also: (Saridakis et al., 2020, Abreu et al., 2021, Prasanthan et al., 5 Oct 2024) for comparative statistical extensions and thermodynamic consistency.