Thermodynamic Volume Product in Spherically Symmetric and Axisymmetric   Spacetime
  In this Letter, we have examined the thermodynamic volume products for spherically symmetric and axisymmetric spacetimes in the framework of \emph{extended phase space}. Such volume products usually formulated in terms of the outer horizon~(${\cal H}^{+}$) and the inner horizon~(${\cal H}^{-}$) of black hole ~ (BH) spacetime. Besides volume product, the other thermodynamic formulations like \emph{volume sum, volume minus and volume division} are considered for a wide variety of spherically symmetric spacetime and axisymmetric spacetimes. Like area~(or entropy) product of multihorizons, the mass-independent~(universal) feature of volume products are sometimes also \emph{fail}. In particular for a spherically symmetric AdS spacetimes the simple thermodynamic volume product of ${\cal H}^{\pm}$ is not mass-independent. In this case, more complicated combinations of outer and inner horizon volume products are indeed mass-independent. For a particular class of spherically symmetric cases i.e. Reissner Nordstr\"om BH of Einstein gravity and Kehagias-Sfetsos BH of Ho\v{r}ava Lifshitz gravity, the thermodynamic volume products of ${\cal H}^{\pm}$ is indeed \emph{universal}. For axisymmetric class of BH spacetime in Einstein gravity all the combinations are \emph{mass-dependent}. There has been no chance to formulate any combinations of volume product relation is to be mass-independent. Interestingly, \emph{only the rotating BTZ black hole} in 3D provides the volume product formula is mass-independent i.e. \emph{universal} and hence it is quantized. 
Parthapratim Pradhan
gr-qc hep-th