A Local First Law of Gravity

We find a (quasi-)local first law of thermodynamics, $\Delta E = T \Delta S - W$, connecting gravitational entropy, $S$, with matter energy and work. For Einstein gravity $S$ is the Bekenstein-Hawking entropy, while for general theories of gravity $S$ is the Wald entropy, evaluated on the stretched future light cone of any point in an arbitrary spacetime, not necessarily containing a black hole. The equation can be written as $\rho \Delta V = T \Delta S - p \Delta V$ by regarding the energy-momentum tensor as that of a fluid.