Entanglement entropy obeys a `first law', much like the ordinary first law of thermodynamics. In any CFT with a semiclassical holographic dual, we show that the first law of entanglement entropy for ball-shaped regions in the CFT is equivalent to the linearized gravitational equation of motion in the bulk, including arbitrary higher curvature corrections. The bulk equations of motion imply the first law of entanglement in the dual CFT, and conversely, the first law can be used to reconstruct the linearized bulk geometry solely from knowledge of the entropy function. The usual Ryu-Takayanagi formula $S={\cal A} /(4\G)$ leads to the linearized Einstein equations, whereas the Wald entropy functional leads to the equations for higher curvature gravity.
Additionally, the first law can be used to derive the holographic dictionary for the stress tensor. In particular, this approach provides a simple alternative to holographic renormalization for computing its expectation value in arbitrary higher derivative gravitational theories. It also carries holographic information about the additional operators in the boundary theory which couple to the bulk metric in higher curvature gravity.
