Holographic Pressure and Extensivity of Rotating Black Holes at Finite Cutoff

Abstract: We investigate the quasi-local thermodynamics of rotating Kerr-AdS black holes enclosed by a finite timelike boundary (cavity). Extending recent work on static systems, we define the holographic pressure and volume via the trace of the Brown-York stress tensor on the cutoff surface. We demonstrate that the inclusion of angular momentum introduces a momentum flux term at the boundary, requiring a generalized first law: $dE = T_{\mathrm{loc}}dS + Ω_{\mathrm{loc}}dJ - \mathcal{P}d\mathcal{A}$. We derive the explicit expressions for these thermodynamic conjugates and analyze the equation of state. Crucially, we examine the extensivity of the system in the large-size limit. We find that while small rotating black holes exhibit non-extensive behavior typical of self-gravitating systems, large Kerr-AdS black holes recover extensivity, behaving effectively as a thermal fluid on the boundary. This result strengthens the holographic interpretation of the cutoff surface as the domain of a dual field theory.