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 λ ∼ C − 1 , or equivalently an “Einstein entropy” that uniquely preserves Einstein's equations for arbitrary C . In both cases, the Unruh temperature is modified to T mod = ℏ κ 2 π 1 + S C , 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, emergent gravity.