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The Einstein equations for generalized theories of gravity and the thermodynamic relation $δQ = T δS$ are equivalent

We show that the equations of motion of generalized theories of gravity are equivalent to the thermodynamic relation $δQ = T δS$. Our proof relies on extending previous arguments by using a more general definition of the Noether charge entropy. We have thus completed the implementation of Jacobson's proposal to express Einstein's equations as a thermodynamic equation of state. Additionally, we find that the Noether charge entropy obeys the second law of thermodynamics if the matter energy momentum tensor obeys the null energy condition. Our results support the idea that gravitation on a macroscopic scale is a manifestation of the thermodynamics of the vacuum.