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We show that the equations of motion of generalized theories of gravity areequivalent to the thermodynamic relation $\delta Q = T \delta S$. Our proofrelies on extending previous arguments by using a more general definition ofthe Noether charge entropy. We have thus completed the implementation ofJacobson's proposal to express Einstein's equations as a thermodynamic equationof state. Additionally, we find that the Noether charge entropy obeys thesecond law of thermodynamics if the matter energy momentum tensor obeys thenull energy condition. Our results support the idea that gravitation on amacroscopic scale is a manifestation of the thermodynamics of the vacuum.

The Einstein equations for generalized theories of gravity and the thermodynamic relation $\delta Q = T \delta S$ are equivalent