Mass and thermodynamic volume in Lifshitz spacetimes

We examine the concept of black hole thermodynamic volume and its consistency with thermodynamic mass in spacetimes that are not asymptotically flat but instead have anisotropic Lifshitz scaling symmetry. We find that the generalized Smarr relation in anti-de Sitter space-extended to include a pressure-volume term-holds here as well, and that there exists a definition of thermodynamic mass and thermodynamic volume that satisfy both this relation and the first law of thermodynamics. We compare the thermodynamic mass with other known quantities such as Arnowitt-Deser-Misner, Brown-York, and Hollands-Ishibashi-Marolf masses. We also conjecture methods for obtaining a thermodynamic mass where there is ambiguity due to the cosmological constant length scale depending on the horizon radius length scale.