This paper studies the thermal properties of charged black holes in anti-de Sitter (AdS) space-time, focusing on their thermodynamic phase structures and field theory interpretations via holography. The black holes are solutions of Einstein–Maxwell truncations of gauged supergravities, arising from rotation in the transverse space. These systems exhibit rich thermodynamic behavior, including classic critical phenomena such as those seen in the van der Waals–Maxwell liquid–gas system. The phase structures are controlled by the "cusp" and "swallowtail" shapes from catastrophe theory, and all thermodynamics is consistent with field theory interpretations. The black holes are studied in arbitrary dimensions (greater than 3), with particular interest in AdS$_4$, AdS$_5$, and AdS$_7$. These correspond to field theories on the world volumes of D3-branes, M2-branes, and M5-branes, respectively. The thermodynamic ensembles considered are the fixed potential and fixed charge ensembles. In the fixed potential ensemble, the background is AdS with a constant potential, while in the fixed charge ensemble, the background is an extremal black hole with that charge. The phase structures in both ensembles are analyzed, revealing interesting behaviors such as the presence of multiple branches of black hole solutions. At intermediate temperatures, the presence of charge allows new branches of solutions to modify the phase structure, leading to complex behaviors. The extremal black hole, even at zero temperature, has non-zero entropy, implying that there must be many states available to the field theory to generate this entropy. The study also reveals that the thermodynamic behavior of these black holes can be mapped to the van der Waals–Maxwell liquid–gas system, with the free energy surface resembling the classic "swallowtail" catastrophe. This suggests a deep connection between the thermodynamics of these black holes and the phase structures of field theories, highlighting the universality of catastrophe theory in describing these systems. The paper also discusses the implications of these results for holography, showing that the thermodynamics of these black holes can be interpreted in terms of field theories, with the dual field theories being conformal field theories on the world volumes of branes. The results are consistent with the AdS/CFT correspondence, and the phase structures observed are consistent with the behavior of field theories in the presence of a global $ U(1) $ current. The study provides new insights into the thermodynamics of these systems and their holographic interpretations.This paper studies the thermal properties of charged black holes in anti-de Sitter (AdS) space-time, focusing on their thermodynamic phase structures and field theory interpretations via holography. The black holes are solutions of Einstein–Maxwell truncations of gauged supergravities, arising from rotation in the transverse space. These systems exhibit rich thermodynamic behavior, including classic critical phenomena such as those seen in the van der Waals–Maxwell liquid–gas system. The phase structures are controlled by the "cusp" and "swallowtail" shapes from catastrophe theory, and all thermodynamics is consistent with field theory interpretations. The black holes are studied in arbitrary dimensions (greater than 3), with particular interest in AdS$_4$, AdS$_5$, and AdS$_7$. These correspond to field theories on the world volumes of D3-branes, M2-branes, and M5-branes, respectively. The thermodynamic ensembles considered are the fixed potential and fixed charge ensembles. In the fixed potential ensemble, the background is AdS with a constant potential, while in the fixed charge ensemble, the background is an extremal black hole with that charge. The phase structures in both ensembles are analyzed, revealing interesting behaviors such as the presence of multiple branches of black hole solutions. At intermediate temperatures, the presence of charge allows new branches of solutions to modify the phase structure, leading to complex behaviors. The extremal black hole, even at zero temperature, has non-zero entropy, implying that there must be many states available to the field theory to generate this entropy. The study also reveals that the thermodynamic behavior of these black holes can be mapped to the van der Waals–Maxwell liquid–gas system, with the free energy surface resembling the classic "swallowtail" catastrophe. This suggests a deep connection between the thermodynamics of these black holes and the phase structures of field theories, highlighting the universality of catastrophe theory in describing these systems. The paper also discusses the implications of these results for holography, showing that the thermodynamics of these black holes can be interpreted in terms of field theories, with the dual field theories being conformal field theories on the world volumes of branes. The results are consistent with the AdS/CFT correspondence, and the phase structures observed are consistent with the behavior of field theories in the presence of a global $ U(1) $ current. The study provides new insights into the thermodynamics of these systems and their holographic interpretations.