  In this work it is shown that the thermodynamics of regular black holes with
a cosmological horizon, which are solutions of Lovelock gravity, determines
that they must evolve either into a state where the black hole and cosmological
horizons have reached thermal equilibrium or into an extreme black hole
geometry where the black hole and cosmological horizons have merged. This
differs from the behavior of Schwarzschild de Sitter geometry which evolves
into a de Sitter space, the ground state of the space of solutions. This occurs
due to a phase transition of the heat capacity of the black hole horizon. To
perform that analysis it is shown that at each horizon a local first law of
thermodynamics can be obtained from the gravitational equations.
