First-order thermodynamics of Horndeski gravity

We extend to the Horndeski realm irreversible thermodynamics description of gravity previously studied in "first generation" scalar-tensor theories. identify a subclass theories as an out-of--equilibrium state, while general relativity corresponds equilibrium state. In this context, we effective heat current, "temperature gravity", and shear viscosity space The identification is accomplished by recasting field equations Einstein with dissipative fluid, following Eckart's first-order thermodynamics.