Which rotating black hole thermodynamics is physically correct?

Kerr-anti-de Sitter black holes admit many apparently valid thermodynamic descriptions, each satisfying the first law and correct scaling — an embarrassment of choices with no obvious selection principle. By applying the quantum statistical relation from the Euclidean path-integral formalism, the authors show that most descriptions are eliminated, leaving a subclass that smoothly recovers Schwarzschild-AdS and ordinary Kerr thermodynamics in the appropriate limits. They further prove that two special descriptions are unique: the one tied to a frame co-rotating with infinity, and the one where thermodynamic and geometric volumes agree. The work clarifies the observer-dependence of black hole thermodynamic quantities and gives a principled way to choose among competing descriptions.