Researchers have formulated a new thermodynamic description for a complex class of black holes, the Kerr-Newman-NUT-AdS$_4$. This formulation introduces the NUT charge parameters, which are not traditionally additional metric parameters, as thermodynamic response variables. Specifically, two "secondary hairs" are defined: a rotation-like variable $J_n = mn/K^2$ and a charge-like variable $N = n/\sqrt{K}$. These, along with the electric charge, pressure, angular momentum, and string tensions, allow for a more complete description of the black hole's thermodynamic state.

The study has achieved a compact formula for the squared mass, of the Christodoulou-Ruffini type, which describes the thermodynamic state of these black holes. By differentiating this equation of state, expressions for the horizon temperature, angular velocities, electric potential, NUT potential, thermodynamic volume, and thermodynamic lengths are obtained. The results algebraically verify the first law of thermodynamics and the Smarr relation, confirming the internal consistency of the new formulation.

This research also explores alternative parameterizations for the NUT charge and clarifies how the choice of thermodynamic volume is linked to the specific NUT sector considered. This work provides a controlled example of how a state space for an AdS black hole can be selected when the consistency of the first law alone is not sufficient to uniquely define it. The advance is significant for understanding the thermodynamics of black holes in anti-de Sitter spaces, which are relevant in the context of the AdS/CFT correspondence.