Exact first law for AdS black holes with anisotropic matter

Determine the exact form of the first law of black hole thermodynamics for the static, spherically symmetric anti–de Sitter black holes constructed in this work that coexist with an anisotropic matter field and metric function f(r) = k − 2M/r + G(r) − (Λ/3) r^2 (where G(r) encodes the anisotropic matter parameters v2 and vc). Identify the correct set of thermodynamic variables and their conjugate potentials that yield a consistent first law for this system.

Background

The paper constructs static AdS black holes dressed by a specific anisotropic matter field and analyzes their thermodynamics and phase structure. Because a recognized action for the fluid-like anisotropic matter is not available, a canonical derivation of thermodynamic relations is subtle.

In Section 3.3, the authors obtain a Smarr-type relation from the horizon condition and then propose a differential first law with newly introduced conjugate pairs (Φ1, D1) and (Φ2, D2) alongside the pressure–volume term for Λ. However, they explicitly state that the exact form of the first law for these matter-dressed black holes remains unknown, motivating a precise determination of the correct thermodynamic variables and potentials.