  One of the striking features of general relativity is that the Einstein
equation is implied by the Clausius relation imposed on a small patch of
locally constructed causal horizon. Extension of this thermodynamic derivation
of the field equation to more general theories of gravity has been attempted
many times in the last two decades. In particular, equations of motion for
minimally coupled higher curvature theories of gravity, but without the
derivatives of curvature, have previously been derived using a thermodynamic
reasoning. In that derivation the horizon slices were endowed with an entropy
density whose form resembles that of the Noether charge for diffeomorphisms,
and was dubbed the Noetheresque entropy. In this paper, we propose a new
entropy density, closely related to the Noetheresque form, such that the field
equation of any diffeomorphism invariant metric theory of gravity can be
derived by imposing the Clausius relation on a small patch of local causal
horizon.
