Correspondence between energy conservation and energy-momentum tensor conservation in cosmology

The correspondence between the thermodynamic energy equation satisfied by a closed comoving volume and conservation energy-momentum tensor of matter inside is extended to more general system with an arbitrary cosmological horizon heat source. consisting internal could be conserved defining surface on horizons. Therefore, can always consistent for such system. On other hand, from perspective classical thermodynamics, one define effective pressure at guarantee that horizon. These systems satisfy generalized second law thermodynamics under appropriate conditions. definitions are gravity theory nonminimal coupling geometry matter, in which regarded as