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With the objective to generalize previous results found for a handful of explicit solutions, we study the extended thermodynamics of a black brane with minimally coupled scalar hair in D-dimensional asymptotically anti-de Sitter spacetimes. Using Komar integration and the Hamiltonian formalism to calculate the conserved charges, we obtain a Smarr relation that is applicable to a wide variety of solutions and suggests a more general definition of the thermodynamic volume. This volume is found to be proportional to the geometric volume, and a simple prescription is given to calculate the constant of proportionality. Moreover, the method of Hamiltonian perturbations yields an extended first law of thermodynamics for hairy black branes, thus giving a definition for their enthalpy. These results are then verified by applying them to some of the explicit solutions that exist in the literature.

Geometric and thermodynamic volume of hairy black branes