Entanglement equilibrium for higher order gravity

We show that the linearized higher derivative gravitational field equations are equivalent to an equilibrium condition on the entanglement entropy of small spherical regions in vacuum. This extends Jacobson's recent derivation of the Einstein equation using entanglement to include general higher derivative corrections. The corrections are naturally associated with the subleading divergences in the entanglement entropy, which take the form of a Wald entropy evaluated on the entangling surface. Variations of this Wald entropy are related to the field equations through an identity for causal diamonds in maximally symmetric spacetimes, which we derive for arbitrary higher derivative theories. If the variations are taken holding fixed a geometric functional that we call the generalized volume, the identity becomes an equivalence between the linearized constraints and the entanglement equilibrium condition. We note that the fully nonlinear higher curvature equations cannot be derived from the linearized equations applied to small balls, in contrast to the situation encountered in Einstein gravity. The generalized volume is a novel result of this work, and we speculate on its thermodynamic role in the first law of causal diamond mechanics, as well as its possible application to holographic complexity.