Extended Gravity Theories from a Thermodynamic Perspective

Abstract: We extend the thermodynamic derivation of gravity in the Jacobson framework by generalizing the Clausius relation through a nontrivial entropy functional. We show that entropy deformations appear as modifications of the effective gravitational coupling, defining a broad class of modified gravity theories. However, conventional entropy corrections are insufficient to resolve spacetime singularities within this approach. We then propose a new entropy form by incorporating quantum properties at the level of horizon degrees of freedom. Implementing this entropy in the modified gravitational framework, we study its cosmological implications at both early and late times. In the early Universe, the model predicts a nonsingular phase with a finite Hubble parameter, leading to a de Sitter-like inflationary expansion with finite entropy and temperature. At late times, the theory reproduces, at leading order, the effective dynamics of loop quantum cosmology.

• The paper shows that generalized entropy in Jacobson’s thermodynamic framework modifies Einstein’s equations primarily through a running effective gravitational coupling, rather than higher-curvature terms.
• The paper finds that conventional logarithmic, power-law, Tsallis, Barrow, Rényi, and Kaniadakis entropy corrections do not remove cosmological singularities because they lack a nonzero minimum horizon area.
• The paper introduces an oscillator-based entropy with minimal area A₀, producing finite early-time Hubble expansion H² = 4π/A₀ and late-time Friedmann dynamics that match loop quantum cosmology at leading order without a bounce.

Overview

This paper develops a thermodynamic derivation of modified gravity within Jacobson's framework, in which the Einstein field equations emerge from the Clausius relation $\delta Q = T\,dS$ applied to local Rindler horizons (2604.09739). The central move is to replace the Bekenstein–Hawking entropy with a general functional $S_{\text{tot}} = f(S_{\text{BH}})$, showing that any such deformation enters the field equations solely as a rescaling of the effective gravitational coupling. The author then demonstrates that conventional entropy corrections (logarithmic, power-law, Tsallis, Barrow, Rényi, Kaniadakis-type) cannot resolve cosmological singularities in this setting, because they all vanish or diverge as the horizon area shrinks to zero. This motivates a new entropy functional built from quantum harmonic oscillator degrees of freedom on the horizon, featuring an explicit minimal area $A_0$. The resulting cosmology is nonsingular: the Hubble parameter saturates at $H^2_{\mathrm{early}} = 4\pi/A_0$, producing a de Sitter-like inflationary phase without inflaton fields, while at late times the model reproduces the leading-order effective Friedmann equation of loop quantum cosmology (LQC).

Generalized entropy and the structure of modified gravity

The derivation follows Jacobson's original construction closely. The heat flux across a local causal horizon is $$\delta Q = -\kappa \int \lambda\, T_{\mu\nu} k^\mu k^\nu\, d\lambda\, dA$$, the temperature is the Unruh temperature $T = \kappa/2\pi$, and the entropy variation is obtained from the Raychaudhuri equation via $\theta \simeq -\lambda R_{\mu\nu}k^\mu k^\nu$. Promoting the Clausius relation to $\delta Q = f'(S_{\text{BH}})\, T\, \delta A$ and treating $f'(S_{\text{BH}})$ as locally constant over an infinitesimal horizon patch, the author obtains

$$G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{2\pi}{f'(S_{\text{BH}})}\, T_{\mu\nu},$$

with the identification $S_{\text{tot}} = f(S_{\text{BH}})$0. The key structural result is that all entropy deformations of this type are dynamically equivalent to a running gravitational coupling; they do not generate higher-curvature terms or other geometric modifications. The derivation assumes local conservation of the effective stress tensor $S_{\text{tot}} = f(S_{\text{BH}})$1, which is an assumption rather than a consequence of the construction — for non-minimally coupled matter this need not hold, and the paper does not address the conditions under which it does.

This result has a direct implication for the phenomenological literature on entropy-corrected cosmologies: models based on Tsallis, Barrow, Rényi, or Kaniadakis entropies are, at the level of the full field equations, simply variable-$S_{\text{tot}} = f(S_{\text{BH}})$2 theories rather than genuinely new gravitational dynamics.

Why conventional corrections fail to resolve singularities

The paper argues that in a spatially flat FRW spacetime the apparent horizon area $S_{\text{tot}} = f(S_{\text{BH}})$3 vanishes as the scale factor approaches zero. Since standard entropy corrections depend directly on $S_{\text{tot}} = f(S_{\text{BH}})$4 without introducing a fundamental lower bound, curvature and energy density divergences persist. The author contrasts this with loop quantum gravity and string theory, where singularity resolution arises from the full quantum dynamics rather than from entropy corrections alone. This is an honest concession about the scope of the framework: singularity resolution cannot be achieved by merely decorating the area law.

A new entropy from horizon microstates

The proposed remedy decomposes the total entropy into macroscopic and microscopic pieces, $S_{\text{tot}} = f(S_{\text{BH}})$5, modeling the horizon degrees of freedom as $S_{\text{tot}} = f(S_{\text{BH}})$6 independent quantum harmonic oscillators. Counting microstates for excitation number $S_{\text{tot}} = f(S_{\text{BH}})$7 gives $S_{\text{tot}} = f(S_{\text{BH}})$8, which for a two-dimensional horizon ($S_{\text{tot}} = f(S_{\text{BH}})$9) yields $A_0$0 in the macroscopic regime. Assuming holographic extensivity of the excitation energy, $A_0$1, where $A_0$2 is the ground-state minimal area, the statistical definition $A_0$3 gives

$A_0$4

The crucial distinction from ordinary logarithmic corrections is the explicit $A_0$5: the horizon cannot shrink below its ground-state area, providing a UV cutoff absent in standard constructions. The form resembles logarithmic corrections known from loop quantum gravity and entanglement entropy calculations, but the physical origin here is a combinatorial count of oscillator excitations combined with a holographic scaling ansatz — both assumptions that are motivated but not derived from first principles.

The paper candidly notes a pathology: as $A_0$6 the logarithm diverges, so the naive continuum description breaks down near the ground state. Two resolutions are offered — quantizing the area as $A_0$7 with $A_0$8, or imposing $A_0$9 so that only excited states enter the thermodynamic description. In both cases the theory enforces a constraint relating $H^2_{\mathrm{early}} = 4\pi/A_0$0 to $H^2_{\mathrm{early}} = 4\pi/A_0$1, and the minimal configuration carries finite entropy and temperature, analogous to a quantum mechanical ground state.

Cosmological dynamics

Combining the modified field equations with the new entropy in a spatially flat FRW universe (no cosmological constant) yields the generalized Friedmann equation

$H^2_{\mathrm{early}} = 4\pi/A_0$2

which, using $H^2_{\mathrm{early}} = 4\pi/A_0$3, becomes implicit in $H^2_{\mathrm{early}} = 4\pi/A_0$4. Selecting the branch consistent with the late-time limit $H^2_{\mathrm{early}} = 4\pi/A_0$5 gives

$H^2_{\mathrm{early}} = 4\pi/A_0$6

Early universe. For $H^2_{\mathrm{early}} = 4\pi/A_0$7, $H^2_{\mathrm{early}} = 4\pi/A_0$8 and the Hubble parameter saturates:

$H^2_{\mathrm{early}} = 4\pi/A_0$9

a finite constant corresponding to de Sitter expansion $$\delta Q = -\kappa \int \lambda\, T_{\mu\nu} k^\mu k^\nu\, d\lambda\, dA$$0. This is the paper's strongest claim: inflation emerges naturally from the entropy structure alone, with no additional scalar field, and the initial singularity is replaced by a regular phase with finite entropy $$\delta Q = -\kappa \int \lambda\, T_{\mu\nu} k^\mu k^\nu\, d\lambda\, dA$$1 and finite temperature $$\delta Q = -\kappa \int \lambda\, T_{\mu\nu} k^\mu k^\nu\, d\lambda\, dA$$2. Consistency between the early-time entropy and the full functional fixes the horizon area to $$\delta Q = -\kappa \int \lambda\, T_{\mu\nu} k^\mu k^\nu\, d\lambda\, dA$$3 via the Lambert function, confirming $$\delta Q = -\kappa \int \lambda\, T_{\mu\nu} k^\mu k^\nu\, d\lambda\, dA$$4 strictly — the ground state is never dynamically reached. In the quantized picture, consistency imposes $$\delta Q = -\kappa \int \lambda\, T_{\mu\nu} k^\mu k^\nu\, d\lambda\, dA$$5 with $$\delta Q = -\kappa \int \lambda\, T_{\mu\nu} k^\mu k^\nu\, d\lambda\, dA$$6, which bounds the allowed quantum number as $$\delta Q = -\kappa \int \lambda\, T_{\mu\nu} k^\mu k^\nu\, d\lambda\, dA$$7.

Late times. Expanding for small $$\delta Q = -\kappa \int \lambda\, T_{\mu\nu} k^\mu k^\nu\, d\lambda\, dA$$8,

$$\delta Q = -\kappa \int \lambda\, T_{\mu\nu} k^\mu k^\nu\, d\lambda\, dA$$9

which coincides in form with the effective Friedmann equation of LQC, with $T = \kappa/2\pi$0 playing the role of the critical density. The author is explicit that this correspondence holds only at leading order in the low-energy expansion and does not extend to the full dynamical structure. More importantly, the ultraviolet behavior differs fundamentally: whereas LQC predicts a nonsingular bounce connecting contraction to expansion, the present model has $T = \kappa/2\pi$1 at high density, so there is no bounce — instead, the universe emerges into an inflationary de Sitter phase. The two frameworks therefore share infrared phenomenology but predict qualitatively different Planck-era physics, a distinction that could in principle be probed through primordial perturbation spectra.

Limitations and open questions

Several caveats bear directly on the results. First, the microscopic entropy construction rests on two modeling assumptions: that horizon degrees of freedom behave as independent harmonic oscillators, and that their excitation energy scales linearly with $T = \kappa/2\pi$2. Neither is derived from an underlying quantum gravity theory. Second, the divergence of $T = \kappa/2\pi$3 at $T = \kappa/2\pi$4 requires an ad hoc prescription (area quantization or exclusion of the ground state), and the choice between these is left open. Third, the conservation of $T = \kappa/2\pi$5 is assumed rather than established, and the status of matter coupling in the variable-$T = \kappa/2\pi$6 sector is unexamined. Fourth, the parameter $T = \kappa/2\pi$7 (equivalently $T = \kappa/2\pi$8) is not fixed by observation; matching the LQC critical density would constrain $T = \kappa/2\pi$9, but no numerical estimate or observational fit is provided. Finally, the analysis is restricted to the homogeneous, isotropic background: black hole physics, cosmological perturbations, structure formation, and stability of the de Sitter phase remain unexplored, as the author acknowledges.

Conclusion

The paper establishes that, within the Jacobson framework, generalized entropy functionals map one-to-one onto effective gravitational couplings, thereby classifying a broad family of entropy-based modified gravity theories as variable-$\theta \simeq -\lambda R_{\mu\nu}k^\mu k^\nu$0 models. It shows that conventional entropy corrections cannot cure the initial singularity, and proposes a minimal-area entropy whose ground-state structure yields a nonsingular cosmology: a finite-$\theta \simeq -\lambda R_{\mu\nu}k^\mu k^\nu$1 de Sitter-like emergent phase in the early universe and LQC-like leading-order dynamics at late times, without a bounce. The framework's predictive content now hinges on fixing $\theta \simeq -\lambda R_{\mu\nu}k^\mu k^\nu$2 observationally and extending the analysis beyond the background FRW evolution.