What is the correct definition of entropy for general relativistic field theory?

Abstract: Recently, the author and collaborators proposed a method to construct a new conserved charge different from the Noether one for general relativistic field theory on curved space-time with energy-momentum tensor covariantly conserved, and that the new conserved charge describes the entropy of the system. As concrete evidence of the proposals, it was shown that such a new conserved charge indeed exists for several classic gravitational systems, and that the proposed entropy density computed in them satisfies the local Euler's relation and the first law of thermodynamics concurrently and non-perturbatively with respect to the Newton constant. These developments are reviewed including brief discussions of physical implications of the local thermodynamic relations.