Thermodynamics of rotating Bardeen black holes: Phase transitions and thermodynamics volume

We investigate the thermodynamical properties of the rotating Bardeen black holes characterized by mass $m$, the spin parameter $a$, and the magnetic charge $g$. We calculate exact expressions of the Hawking temperature, mass, entropy, and heat capacity. The black hole mass is minimum at radius ${r}_{+}^{E}$, where both the heat capacity and temperature vanish with a stable remnant. Also, there exists a critical radius ${r}_{+}^{C}$ of multiple orders, where the heat capacity diverges, suggesting that the black hole is thermodynamically stable in the range ${r}_{+}^{E}<{r}_{+}<{r}_{+}^{C}$. We also analyze the extended phase space thermodynamics of the rotating Bardeen--anti-de Sitter black holes. Treating the cosmological constant $\mathrm{\ensuremath{\Lambda}}$ and charge $g$ as thermodynamic variables, we derive the generalized first law in the extended phase space to study the critical phenomena of the black holes. The Ehrenfest scheme for the $P\ensuremath{-}V$ criticality of the rotating Bardeen black holes in anti-de Sitter spaces is examined. The Clausius-Clapeyron-Ehrenfest equations confirming the second-order phase transitions of the van der Waals fluid is no longer valid.

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