Explicit and universal formula for thermodynamic volume in extended black hole thermodynamics

Abstract: In extended black hole thermodynamics, the cosmological constant and other couplings are treated as thermodynamic variables, yielding a first law $$\tildeδM=T\tildeδS+Ω\tildeδJ+\mathcal{V}\tildeδP+\cdots$$, where $P\equiv -\fracΛ{8π}$. A long-standing conceptual gap in this framework is that, unlike $M$, $T$, $S$, $Ω$, and $J$, the thermodynamic volume $\mathcal{V}$ lacks a first-principles definition and can only be deduced from other thermodynamic quantities. This deficiency indicates that the underlying origin of $\mathcal{V}$ has remained poorly understood. In this work, we resolve this issue and provide an explicit universal formula for $\mathcal{V}$. We demonstrate that it universally decomposes into two contributions, one arising from the explicit dependence of the action on the couplings and the other from the response of the fundamental dynamical fields. This clarifies the physical meaning of thermodynamic volume and places it on the same footing as other intrinsic thermodynamic quantities.