Rotating black hole in $f(R)$ theory

In general, the field equation of $f(R)$ gravitational theory is very intricate, and therefore, it is not an easy task to derive analytical solutions. We consider rotating black hole spacetime four-dimensional in the $f(R)$ gravitational theory and derive a novel black hole solution. This black hole reduced to the one presented in \cite{Nashed:2020mnp} when the rotation parameter, $\Omega$, vanishes. We study the physical properties of this black hole by writing its line element and show that it asymptotically behaves as the AdS/dS spacetime. Moreover, we derive the values of various invariants finding that they do possess the central singularity, and show that our black hole has a strong singularity compared with the black hole of the Einstein general relativity (GR). We calculate several thermodynamical quantities and show that we have two horizons, the inner and outer Cauchy horizons in contrast to GR. From the calculations of thermodynamics, we show that the outer Cauchy horizon gives satisfactory results for the Hawking temperature, entropy, and quasi-local energy. Moreover, we show that our black hole has a positive value of the Gibbs free energy which means that it is a stable one. Finally, we derive the stability condition analytically and graphically using the geodesic deviation method.