Microscopic description of thermodynamic volume in extended black hole thermodynamics

Using the fact that theories of gravity with asymptotically three-dimensional anti-de Sitter geometries have dual descriptions as two-dimensional conformal field theories (CFTs), we present the first study in field theory of the thermodynamic volume of various black hole solutions. We explain in general two-dimensional CFI' terms why the presence of a thermodynamic volume can render certain black hole solutions superentropic. Superentropicity simply results from the fact that the Cardy formula, which gives the gravitational Bekenstein-Hawking entropy, can overcount the CFI' entropy. The examples of charged Banados, Teitelbiom, and Zanelli (BTZ) black holes and generalized exotic BTZ black holes are described. These observations help explain why the specific heat at constant volume can signal the instability of such solutions, as recently conjectured.