The thermodynamics for Kerr-AdS black hole in four dimensions is revisited using the recently proposed restricted phase space formalism, which includes the central charge C of the dual CFT and the chemical potential $$\mu $$ μ , but excludes the pressure and the conjugate volume, as thermodynamic variables. The Euler relation holds automatically, and the first order homogeneity of the mass and the zeroth order homogeneity of the intensive variables are made explicit. Thermodynamic processes involving each pair of conjugate variables are studied in some detail, with emphasis on the scaling properties of the equations of states. It turns out that the thermodynamic behavior of the Kerr-AdS black hole is very similar to that of the RN-AdS black hole studied earlier. In particular, it is found that, there is a first order supercritical phase equilibrium in the $$T-S$$ T - S processes at fixed nonvanishing angular momentum, while at vanishing angular momentum or at fixed angular velocities, there is always a non-equilibrium transition from a small unstable black hole state to a large stable black hole state. Moreover, there is a Hawking–Page phase transition in the $$\mu -C$$ μ - C processes. Due to the complicatedness of the Kerr metric, the exact critical point and the Hawking–Page temperature are worked out explicitly only in the slow rotating limit, however the characteristic thermodynamic properties do not rely on the slow rotating approximation.

A new formalism for thermodynamics of AdS black holes called the {\em restricted phase space thermodynamics} (RPST) is proposed. The construction is based on top of Visser's holographic thermodynamics, but with the AdS radius fixed as a constant. Thus the RPST is free of the $(P,V)$ variables but inherits the central charge and chemical potential as a new pair of conjugate thermodynamic variables. In this formalism, the Euler relation and the Gibbs-Duhem equation hold simultaneously with the first law of black hole thermodynamics, which guarantee the appropriate homogeneous behaviors for the black hole mass and the intensive variables. The formalism is checked in detail in the example case of 4-dimensional RN-AdS black hole in Einstein-Maxwell theory, in which some interesting thermodynamic behaviors are revealed.

The recently proposed restricted phase space thermodynamics is shown to be applicable to a large class of higher dimensional higher curvature gravity models coupled to Maxwell field, which are known as black hole scan models and are labeled by the spacetime dimension d and the highest order k of the Lanczos-Lovelock densities appearing in the action. Three typical example cases with (d,k)=(5,1),(5,2) and (6,2) are chosen as example cases and studied in some detail. These cases are representatives of Einstein-Hilbert, Chern-Simons and Born-Infield like gravity models. Our study indicates that the Einstein-Hilbert and Born-Infield like gravity models have similar thermodynamic behaviors, e.g., the existence of isocharge T−S phase transitions with the same critical exponents, the existence of isovoltage T−S transitions and the Hawking-Page like transitions, and the similar high temperature asymptotic behaviors for the isocharge heat capacities, etc. However, the Chern-Simons like (5,2)-model behaves quite differently. Neither isocharge nor isovoltage T−S transitions could occur and no Hawking-Page like transition is allowed. This seems to indicate that the Einstein-Hilbert and Born-Infield like models belong to the same universality class while the Chern-Simons like models do not.