F(R) gravity equation of state

The $\mathbf{f}(R)$ gravity field equations are derived as an equation of state of local space-time thermodynamics. Jacobson’s arguments are non-trivially extended, by means of a more general definition of local entropy, for which Wald’s definition of dynamic black hole entropy is used, as well as the concept of an effective Newton constant for graviton exchange, recently appeared in the literature.