Beyond comoving volume: horizon flux and matter creation in modified cosmology from the unified first law of thermodynamics

Abstract We explore the derivation of the Friedmann equations from a thermodynamic perspective, applying the unified first law of thermodynamics to the apparent horizon of a flat Friedmann–Lemaître–Robertson–Walker (FLRW) universe. We extend this framework to incorporate gravitationally induced particle creation, treating the region enclosed by the apparent horizon as an open thermodynamic system. A crucial aspect of our analysis is the recognition that the apparent horizon volume is not comoving; this requires consistent accounting of particle exchange across the moving boundary. We demonstrate that the evolution of the particle number, and explicitly the matter entropy, can be decomposed into two distinct physical contributions: genuine bulk particle production and a net flux induced by the dynamics of the horizon itself. Finally, we derive the Generalized Second Law (GSL) in this setting, showing transparently how the total entropy budget is balanced by horizon thermodynamics, bulk creation, and boundary fluxes.