This paper explains a puzzle about energy and volume in rotating black holes that live in anti‑de Sitter space (a curved spacetime with a negative cosmological constant). Earlier work produced two different energy-like quantities for Kerr–anti‑de Sitter black holes. One, called E, fits the usual first law of black hole thermodynamics. The other, called F, obeys a related scaling law (a Smarr relation) in which the volume that appears is a simple geometric or “vector” volume. The author asks why E is needed for the first law while the geometric volume appears naturally in the F–Smarr law.

To answer this, the author builds a conserved‑charge framework adapted to Kerr–Schild representations of the spacetime. He follows methods of Barnich and Compère to define a (D−2)-form I_χ for any symmetry (Killing vector) χ. Integrating that form over a surface around the black hole gives a conserved quantity H^I_χ. When χ is the asymptotically nonrotating, hypersurface‑orthogonal Killing vector ξ, H^I_ξ equals E. When χ is a different vector β, related to the divergence of the principal conformal Killing–Yano tensor (a special geometric object denoted h), H^I_β equals F. The Kerr–Schild decomposition is used to separate the black hole spacetime into a background anti‑de Sitter metric plus a perturbation, which makes these charge integrals tractable.

At a conceptual level the paper shows why the first law works for E but not for F. The first law with a variable cosmological constant (interpreted as pressure P) and its conjugate thermodynamic volume V_th requires that both the background anti‑de Sitter metric and the Killing vector used to build the charge have components that do not change under the allowed variations. That is true for ξ but not for β. Because β’s components depend on structures that vary with the black hole parameters, H^I_β does not yield quantities that satisfy the first law. Separately, algebraic simplifications tied to the principal conformal Killing–Yano tensor h make the vector or geometric volume V_geo (which the author shows equals the vector volume V_C) appear naturally in the Smarr relation associated with β.