Eur. Phys. J. C (2025) 85: 1032
https://doi.org/10.1140/epjc/s10052-025-14722-9

Regular Article - Theoretical Physics

Generalized free energy of black holes and nucleated bubbles in the gauged Kaluza–Klein theory

¹ Phenikaa Institute for Advanced Study, Phenikaa University, Nguyen Trac, Duong Noi, 100000, Hanoi, Vietnam
² Institute of Theoretical and Applied Research, Duy Tan University, 100000, Hanoi, Vietnam
³ School of Engineering and Technology, Duy Tan University, 550000, Da Nang, Vietnam

^a This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 16 June 2025
Accepted: 30 August 2025
Published online: 19 September 2025

Abstract

We explore the phase emergence and corresponding phase transitions of charged dilaton $\text{AdS}$ black holes in the gauged Kaluza–Klein theory in the context of the generalized free energy. The black hole solutions are asymptotic to the AdS$_D^{}$ geometry, and one can obtain them from the dimensional reduction of gauged supergravities on the compact internal manifolds, leading to the restriction as $4\le D\le 7$. In the canonical ensemble, we classify the black hole thermodynamics into three different classes as follows: (i) $D=4$, (ii) $D=5$, and (iii) $D=6,7$. In the grand canonical ensemble, the black hole thermodynamics is independent on the number of spacetime dimensions and the pressure, but it depends on the chemical potential $\mathrm{\Phi }$. As a result, we classify the black hole thermodynamics as follows: (i) $\mathrm{\Phi }<1$, (ii) $\mathrm{\Phi }>1$, and (iii) $\mathrm{\Phi }=1$. In addition, we show that the asymptotically AdS solutions can provide a scenario where our Universe lives on a nucleated bubble mediating the decay of unstable AdS space. We point out a minimum value for the scale factor in the case of a neutral bubble by which the initial curvature singularity can be removed. We obtain a dS universe corresponding to the late-time evolution of the nucleated bubble, avoiding the constraint of the dS conjecture.