Charged spherically symmetric Taub–NUT black hole solutions in $f(R)$ gravity

$f(R)$ theory is a modification of Einstein’s general relativity which has provided many interesting results in cosmology and astrophysics. To derive a black hole solution in this theory is difficult due to the fact that it contains fourth-order differential equations. In this study, we use the first reliable deviation from general relativity which is given by the quadratic form of $f(R)=R+\beta R^2$, where $\beta$ is a dimensional parameter. We calculate the energy conditions of charged black holes and show that they are all satisfied for the Taub–NUT spacetime. Finally, we study some thermodynamic quantities such as entropy, temperature, specific heat, and Gibbs free energy. The calculations of heat capacity and free energy show that the charged Taub–NUT black hole has positive values, which means that it has thermal stability.