On the properties of a Static and a Stationary Charged Black Hole in $f(R)$ Gravity

In a recent paper, Nashed and Capozziello presented a new class of charged, spherically symmetric black hole solutions of $f(R)$ gravity with an asymptotic flat or (anti-)de Sitter behavior. These metrics depend on a dimensional parameter $\alpha$ and are interesting because they cannot reduce to general relativity solutions. In this paper, we present a corrected study of their physical and thermodynamic properties and generalize these solutions to obtain a new set of stationary, axisymmetric black holes in the $f(R)$ scenario. Some of our results show that the entropy is always positive within the allowed values of parameter $\alpha$ and due to the well behaved quantities such as the Gibbs free energy, we conclude that there is no such a phase transition as discussed in the work of Nashed and Capozziello. We also study the geodesics in these spacetimes and particularly, the stability of the circular orbits to obtain the radius of the Innermost Stable Circular Orbit.