The First Law of Thermodynamics in AdS5 in the Presence of Conformal Anomalies

In this talk I will present the results that we recently reported in arXiv:2209.09031 (cur- rently under review by JHEP). In this research we proposed a solution to a long-standing problem with the compatibility of the counterterms subtraction method with the first law and generalized Smarr's formula in extended phase-space. We used the counterterms sub- traction method to calculate various thermodynamical quantities for charged rotating black holes in five-dimensional minimal gauged supergravity. Specifically, we analyzed certain issues related to the first law and Smarr's relation in the presence of a conformal anomaly. Among the bulk quantities calculated are the on-shell action, total mass, and angular mo- menta of the solution. All these quantities are consistent with previous calculations made using other methods. For the boundary theory, we calculate the renormalized stress tensor, conformal anomaly, and Casimir energy. Using the Papadimitriou-Skenderis analysis [2], we showed that the mass calculated via the counterterms method satisfies the first law of black hole thermodynamics. To discuss extended thermodynamics, we extended the definition of the thermodynamic volume to cases with conformal anomalies using a procedure similar to that of Papadimitriou-Skenderis. We showed that this volume correctly accounts for extra terms due to boundary metric variation. This shows that the mass and volume calculated using counterterms satisfy Smarr's relation as well as the first law.