Scalar-tensor theories of gravity from a thermodynamic viewpoint

In any diffeomorphism invariant theory of gravity, one can define a Noether charge arising from the invariance of the Lagrangian under diffeomorphisms. We have determined the Noether charge for scalar-tensor theories of gravity, in which case the gravity is mediated by the metric tensor as well as by a scalar degree of freedom. In particular, we demonstrate that the total Noether charge within an appropriate spatial volume can be related to the heat content of the boundary surface. For static spacetimes, in these theories, there exist an ``equipartition" between properly defined bulk and surface degrees of freedom. While the dynamical evolution of spacetime in these theories of scalar-tensor gravity arises due to the departure from the equipartition regime. These results demonstrate that thermodynamical interpretations for gravitational theories transcend Einstein and Lovelock theories of gravity, holding true for theories with additional scalar degrees of freedom as well. Moreover, they hold in both the Jordan and the Einstein frames. However, it turns out that there are two dynamically equivalent representations of the scalar-tensor theory in the Jordan frame, differing by total derivatives in the action, which are thermodynamically inequivalent. This depicts the importance of having a thermodynamic description, which can be used in distinguishing various dynamically equivalent representations of gravity theories beyond Einstein.