Modified Unruh Thermodynamics in Emergent Gravity: Finite Heat Capacity and Rényi Entropy

We show that Jacobson’s thermodynamic derivation of Einstein’s equations remains valid when local Rindler horizons are modeled as finite heat-capacity systems, resolving the infinite-bath assumption of Unruh thermodynamics. The horizon entropy then takes the form of Rényi entropy with nonextensivity parameter $\lambda\sim C^{-1}$, or equivalently a new “Einstein entropy” that uniquely preserves Einstein’s equations for arbitrary $C$. In both cases the Unruh temperature is modified to

\begin{equation*}

T_\text{mod}=\frac{\hbar\kappa}{2\pi}\left(1+\frac{S}{C}\right),

\end{equation*}

establishing a universal link between finite-capacity thermodynamics and generalized entropies. We further derive a corrected scalar Einstein equation with an upper bound on horizon energy flux, suggesting testable signatures in heavy-ion collisions, spin-polarization experiments, and analog gravity.