Thermodynamic gravity with non-extensive horizon entropies and topological calibration

Abstract We revisit Jacobson’s thermodynamic derivation of gravity for generalized, non-extensive horizon entropies and develop a consistent framework in which modified gravitational dynamics emerge from horizon thermodynamics. By interpreting S ( A ) as a coarse-grained entropy entering the thermodynamic constitutive relation, we show that only its local slope at a reference area A ∗ contributes to macroscopic dynamics within a local Rindler-wedge construction. In this setting, the Clausius relation is reformulated as a stationarity condition for a Massieu functional at fixed Unruh temperature, allowing the entropy slope s 0 = ( d S / d A ) | A ∗ to be identified as the parameter controlling the effective gravitational coupling. For entropy linear in area, Einstein’s equations are recovered with G eff = 1 / ( 4 s 0 ) . As a separate consistency check, we demonstrate that a scalar-response extension of the horizon entropy density, supplemented by internal entropy production, correctly reproduces the field equations of f ( R ) gravity. Motivated by group-entropic arguments and generalized thermodynamics, we adopt a power-law entropy S ( A ) = η ( A / 4 G ) δ and study its implications across scales. To remove the ambiguity in the coarse-graining scale, we introduce a Topological Calibration Principle linking the reference area to intrinsic geometric data through the Gauss–Bonnet theorem. For compact cross-sections admitted by the chosen theory or boundary conditions, this yields a topology-dependent effective coupling governed by the Euler characteristic. Consistency requirements constrain deviations from extensivity and generate scale-dependent modifications with potentially observable cosmological signatures. The resulting framework provides a systematic setting for testing non-extensive horizon thermodynamics and its phenomenological implications.