Thermodynamics of spacetime in generally covariant theories of gravitation

It has been shown that the Einstein equation can be derived from the requirement that the First Law δQ = TdS holds for all local Rindler causal horizons through each spacetime point. Here δQ and T are the energy flux and Unruh temperature seen by an accelerating observer just inside the horizon, and the entropy S is one quarter the horizon area in Planck units. In this paper we show that there are problems extending this thermodynamic derivation to include generic diffeomorphism invariant theories of gravitation, where the entropy is the integral of the Noether charge over the horizon. The requirement of thermodynamic equilibrium is demonstrated to be incompatible with the local stationarity of the horizon. In scalar-tensor theories this obstruction can be remedied with a conformal transformation of the metric, but this method fails to work in higher curvature theories. We conclude by using effective field theory to argue that the thermodynamic derivation should only apply in the lowest order regimes of the effective gravitational action.