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. The 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.
