  $f(R)$ theory is a modification of Einstein general relativity which has many
interesting results in cosmology and astrophysics. To derive black hole
solution in this theory is difficult due to the fact that it is 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
the charged black holes and show that all of them are 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 have
positive values which means that it has thermal stability.
