Regular Multi-Horizon Black Holes in Modified Gravity with Non-Linear
  Electrodynamics
  We investigated the regular multi-horizon black holes in the Einstein
gravity, $F(R)$ gravity and the 5 dimensional Gauss-Bonnet gravity, all of them
coupled with non-linear electrodynamics. We presented several explicit examples
of the actions which admit the solutions describing regular black hole
space-time with multi-horizons. 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 multi-horizon 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.
