  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.
