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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>λ</a:mi> <a:mo>∼</a:mo> <a:msup> <a:mi>C</a:mi> <a:mrow> <a:mo>−</a:mo> <a:mn>1</a:mn> </a:mrow> </a:msup> </a:mrow> </a:math> , or equivalently an “Einstein entropy” that uniquely preserves Einstein's equations for arbitrary <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>C</b:mi> </b:math> . In both cases, the Unruh temperature is modified to <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:msub> <c:mi>T</c:mi> <c:mtext>mod</c:mtext> </c:msub> <c:mo>=</c:mo> <c:mfrac> <c:mrow> <c:mi>ℏ</c:mi> <c:mi>κ</c:mi> </c:mrow> <c:mrow> <c:mn>2</c:mn> <c:mi>π</c:mi> </c:mrow> </c:mfrac> <c:mfenced separators="" open="(" close=")"> <c:mn>1</c:mn> <c:mo>+</c:mo> <c:mfrac> <c:mi>S</c:mi> <c:mi>C</c:mi> </c:mfrac> </c:mfenced> </c:mrow> </c:math> , 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.