The on-shell gravitational action and the boundary stress tensor are essential ingredients in the study of black hole thermodynamics. We employ the Hamilton–Jacobi method to calculate the boundary counterterms necessary to remove the divergences and allow the study of the thermodynamics of Einstein–Gauss–Bonnet black holes. PACS numbers: 04.20.Fy, 04.50.Gh, 04.70.Dy, 11.10.Kk

This paper aims to clarify the relation between the second law of thermodynamics and the notion of irreversibility by distinguishing three different meanings of the latter and to study how they figure in three versions of the second law of thermodynamics. A more extensive discussion is given

It is shown that the laws of thermodynamics are extremely robust under generalizations of the form of entropy. Using the Bregman-type relative entropy, the Clausius inequality is proved to be always valid. This implies that thermodynamics is highly universal and does not rule out consistent generalization of the maximum entropy method.

Based on a covariant theory of equilibrium Thermodynamics, a Statistical Relativistic Mechanics is developed for the non-interacting case. Relativistic Thermodynamics and Jüttner Relativistic Distribution Function in a moving frame are obtained by using this covariant theory. A proposal for a Relativistic Statistical Mechanics is exposed for the interacting case.

Experimental observations made upon a thermodynamic system containing a mixture of living matter and inert matter were published in the recent years. They seem incompatible with the usual conception of thermodynamics, but compatible with an extended conception which links thermodynamics and relativity. Nevertheless, they ask an interesting question about the behavior of living matter.

Introducing the notion of thermal entropy density via the first law of thermodynamics and assuming the Einstein equation as an equation of thermal state, we obtain the thermal entropy density of any arbitrary spacetime without assuming a temperature or a horizon. The results confirm that there is a profound connection between gravity and thermodynamics.