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

We explore the phase emergence and corresponding phase transitions of charged dilaton AdS 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 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≤ D≤ 7 4 ≤ D ≤ 7 . In the canonical ensemble, we classify the black hole thermodynamics into three different classes as follows: (i) $$D=4$$ D = 4 , (ii) $$D=5$$ D = 5 , and (iii) $$D=6,7$$ 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 Φ Φ . As a result, we classify the black hole thermodynamics as follows: (i) Φ <1 Φ < 1 , (ii) Φ >1 Φ > 1 , and (iii) Φ =1 Φ = 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.