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The variation of the energy for a gravitational system is directly definedfrom the Hamiltonian field equations of General Relativity. When the variationof the energy is written in a covariant form it splits into two (covariant)contributions: one of them is the Komar energy, while the other is theso-called covariant ADM correction term. When specific boundary conditions areanalyzed one sees that the Komar energy is related to the gravitational heatwhile the ADM correction term plays the role of the Helmholtz free energy.These properties allow to establish, inside a classical geometric framework, aformal analogy between gravitation and the laws governing the evolution of athermodynamic system. The analogy applies to stationary spacetimes admittingmultiple causal horizons as well as to AdS Taub-bolt solutions.Comment: Latex file, 31 pages; one reference and two comments added, misprints  correcte

Boundary conditions, energies and gravitational heat in general relativity (a classical analysis)