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The correspondence between the thermodynamic energy equation satisfied by aclosed co-moving volume and the conservation equation satisfied by theenergy-momentum tensor of the matter inside the co-moving volume is extended toa more general system with an arbitrary cosmological horizon and a heat source.The energy of the system consisting of a cosmological horizon and its internalmatter could be conserved by defining a surface energy on the horizons.Therefore, energy conservation and energy-momentum tensor conservation canalways be consistent for such a system. On the other hand, from the perspectiveof classical thermodynamics, one can define an effective pressure at thecosmological horizon to guarantee that the thermodynamic energy equation insidethe horizon is consistent with the energy-momentum tensor conservation equationof the matter inside the horizon. These systems can satisfy the generalizedsecond law of thermodynamics under appropriate conditions. The definitions ofthe surface energy and the effective pressure are extended to the gravitytheory with non-minimal coupling between geometry and matter, in which geometrycould be regarded as a heat source.

On the correspondence between energy conservation and energy-momentum tensor conservation in cosmology