  We explicitly show that the equations of motion for modified gravity theories
of $F(R)$-gravity, the scalar-Gauss-Bonnet gravity, $F(\mathcal{G})$-gravity
and the non-local gravity are equivalent to the Clausius relation in
thermodynamics. In addition, we discuss the relation between the expression of
the entropy and the contribution from the modified gravity as well as the
matter to the definition of the energy flux (heat).
