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By considering a small correction to the Maxwell field, we show that the resultant black hole solutions (also known as the asymptotically Reissner–Nordström black holes) undergo the reentrant phase transition and can have a novel phase behavior. We also show that such a small nonlinear correction of the Reissner–Nordström black holes has high effects on the phase structure of the solutions. It leads to a new classification in the canonical ensemble of extended phase space providing the values of the nonlinearity parameter α being α⋚4q2/7. We shall study these three classes and investigate deviations from those of the standard Reissner–Nordström solutions. Interestingly, we find that there is the reentrant phase transition for α

In this paper, we first investigate the thermal stability of black holes in conformal Weyl gravity with a comparison with the Schwarzschild black holes. Then, we consider a minimally coupled massive scalar perturbation and calculate the quasinormal modes in asymptotically dS spacetime by employing the sixth order WKB approximation and asymptotic iteration method. The deviations from those of the Schwarzschild-dS solutions are obtained and the possibility of the presence of quasi-resonance modes for Weyl black hole solutions is investigated. Finally, we consider a massless scalar perturbation in the background of asymptotically AdS solutions and calculate the quasinormal modes by using the pseudospectral method. The effects of the free parameter of the theory on the quasinormal modes are studied and deviations from those of the Schwarzschild-AdS black holes are investigated. The imaginary part of quasinormal frequencies in AdS spacetime is the time scale of a thermal state (in the conformal field theory) to approach thermal equilibrium.

Motivated by the interesting non-abelian gauge field, in this paper, we look for the analytical solutions of Yang–Mills theory in the context of gravity's rainbow. Regarding the trace of quantum gravity in black hole thermodynamics, we examine the first law of thermodynamics and also thermal stability in the canonical ensemble. We show that although the rainbow functions and Yang–Mills charge modify the solutions, the first law of thermodynamics is still valid. Based on the phenomenological similarities between the adS black holes and van der Waals liquid/gas systems, we study the critical behavior of the Yang–Mills black holes in the extended phase space thermodynamics. We also investigate the effects of various parameters on thermal instability as well as critical properties by using appropriate figures.

Magnetic branes of Gauss–Bonnet–Maxwell theory in the context of massive gravity is studied in detail. Exact solutions are obtained and their interesting geometrical properties are investigated. It is argued that although these horizonless solutions are free of curvature singularity, they enjoy a cone-like geometry with a conic singularity. In order to investigate the effects of various parameters on the geometry of conic singularity, its corresponding deficit angle is studied. It will be shown that despite the effects of Gauss–Bonnet gravity on the solutions, deficit angle is free of Gauss–Bonnet parameter. On the other hand, the effects of massive gravity, cosmological constant and electrical charge on the deficit angle will be explored. Also, a brief discussion related to possible geometrical phase transition of these topological objects is given.