We derive the gravitational equations of motion of general theories of gravity from thermodynamics applied to a local Rindler horizon through any point in spacetime. Specifically, for a given theory of gravity, we substitute the corresponding Wald entropy into the Clausius relation. Our approach works for all diffeomorphism-invariant theories of gravity in which the Lagrangian is a polynomial in the Riemann tensor.
Parikh, M., & Sarkar, S. (2016). Generalized Einstein’s Equations from Wald Entropy. Entropy, 18(4), 119. doi:10.3390/e18040119
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