  It has previously been shown that the Einstein equation can be derived from
the requirement that the Clausius relation dS = dQ/T hold for all local
acceleration horizons through each spacetime point, where dS is one quarter the
horizon area change in Planck units, and dQ and T are the energy flux across
the horizon and Unruh temperature seen by an accelerating observer just inside
the horizon. Here we show that a curvature correction to the entropy that is
polynomial in the Ricci scalar requires a non-equilibrium treatment. The
corresponding field equation is derived from the entropy balance relation dS
=dQ/T+dS_i, where dS_i is a bulk viscosity entropy production term that we
determine by imposing energy-momentum conservation. Entropy production can also
be included in pure Einstein theory by allowing for shear viscosity of the
horizon.
