Thermodynamics of 5D Charged Rotating Black Holes: A Counterterms Treatment

We show that the calculation of the thermodynamic volume in the bulk theory is sensitive to conformal anomalies in the dual CFT, and present a new way of calculating it when the conformal anomalies do not vanish. This solves the issue where the use of the counterterms subtraction method appears to break the first law and Smarr's relation in extended phase-space. This is shown explicitly in the case of charged rotating black holes in five-dimensional minimal gauged supergravity (Chong et al. in Phys Rev Lett 95:161301, 2005), where we use the counterterms method to study the thermodynamics of the solution. Among the bulk quantities calculated using the counterterms method are the on-shell action, total mass, and angular momenta of the solution. All these quantities are consistent with previous calculations made using other methods. For the boundary CFT, we calculate the renormalized stress tensor, conformal anomaly, and Casimir energy. Using the Papadimitriou-Skenderis analysis (Papadimitriou and Skenderis in J. High Energy Phys. 08:004, 2005), we show that the mass calculated via the counterterms method satisfies the first law of black hole thermodynamics. To discuss extended thermodynamics, we extend the definition of the thermodynamic volume to cases with conformal anomalies using a procedure similar to that of Papadimitriou-Skenderis. We show that this volume correctly accounts for extra terms due to boundary metric variation.