Regular multihorizon black holes in modified gravity with nonlinear electrodynamics

We investigated the regular multihorizon black holes in the Einstein gravity, F (R) gravity, and the 5 dimensional Gauss-Bonnet gravity, all of them coupled with nonlinear electrodynamics. We presented several explicit examples of the actions which admit the solutions describing regular black hole space-time with multihorizons. Thermodynamics of the obtained black hole solutions is studied. The explicit expressions of the temperature, the entropy, the thermodynamical energy and the free energy are obtained. Although the temperature vanishes in the extremal limit where the radii of the two horizons coincide with each other as in the standard multihorizon black hole like the Reissner-Nordstr\"om black hole or the Kerr black hole, the larger temperature corresponds to the larger horizon radius. This is different from the standard black holes, where the larger temperature corresponds to the smaller horizon radius. We also found that the specific heat becomes positive for the large temperature, which is also different from the standard black holes, where the specific heat is negative. It should be also noted that the thermodynamical energy is not identical with the ADM mass. Furthermore in case of the Gauss-Bonnet gravity, it is demonstrated that the entropy can become negative.