Thermodynamic Volume Product in Spherically Symmetric and Axisymmetric Spacetime

We have examined the thermodynamic volume products for spherically symmetric and axisymmetric spacetime in the framework of extended phase space. Such volume products are usually formulated in terms of the outer horizon (H+) and the inner horizon (H-) of black hole (BH) spacetime. Besides volume product, the other thermodynamic formulations like volume sum, volume minus, and volume division are considered for a wide variety of spherically symmetric spacetime and axisymmetric spacetime. Like area (or entropy) product of multihorizons, the mass-independent (universal) features of volume products sometimes also fail. In particular, for a spherically symmetric AdS spacetime, the simple thermodynamic volume product of H± 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 Nordström BH of Einstein gravity and Kehagias-Sfetsos BH of Hořava Lifshitz gravity, the thermodynamic volume products of H± are indeed universal. For axisymmetric class of BH spacetime in Einstein gravity, all the combinations are mass-dependent. There has been no chance to formulate any combinations of volume product relation to be mass-independent. Interestingly, only the rotating BTZ black hole in 3D provides that the volume product formula is mass-independent, i.e., universal, and hence it is quantized.