Refined thermodynamics of black holes with proper conserved charges

Field equations of a classical, geometric, theory of gravity, augmented with some semiclassical considerations strongly suggest that the gravitational field representing a stationary black hole can be simply described with a few thermodynamical coordinates and their conjugates that obey the four laws of thermodynamics plus the Smarr formula and the reverse isoperimetric inequality that bounds the maximum entropy for a given effective volume of space. The thermodynamics of black holes is a promising window to the quantum nature of black holes; hence, it is important to understand all the details of these laws. The identification and the meaning of these thermodynamic coordinates depend on the gravity theory under consideration. For example, the existence of dimensionful coupling constants, such as the cosmological constant, changes the scaling properties of the theory, its solutions, and the laws of thermodynamics. Here we show, using the background Killing charge method, which applies to the black hole solutions of any gravity theory that has a maximally symmetric vacuum, to define the mass and angular momentum, instead of using the Komar mass and the angular momentum, how the thermodynamics of black holes such as the $D$ dimensional Kerr-AdS black holes in cosmological Einstein's theory and the spherically symmetric black holes in the Einstein-Gauss-Bonnet theory changes. We give the effective volume of black holes even without a cosmological constant.