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We investigate static and rotating charged spherically symmetric solutions inthe framework of $f({\cal R})$ gravity, allowing additionally theelectromagnetic sector to depart from linearity. Applying a convenient, dualdescription for the electromagnetic Lagrangian, and using as an example thesquare-root $f({\cal R})$ correction, we solve analytically the involved fieldequations. The obtained solutions belong to two branches, one that contains theKerr-Newman solution of general relativity as a particular limit and one thatarises purely from the gravitational modification. The novel black holesolution has a true central singularity which is hidden behind a horizon,however for particular parameter regions it becomes a naked one. Furthermore,we investigate the thermodynamical properties of the solutions, such as thetemperature, energy, entropy, heat capacity and Gibbs free energy. We extractthe entropy and quasilocal energy positivity conditions, we show thatnegative-temperature, ultracold, black holes are possible, and we show that theobtained solutions are thermodynamically stable for suitable model parameterregions.

New rotating black holes in non-linear Maxwell $f({\cal R})$ gravity