The instabilities and (anti)-evaporation of Schwarzschild-de Sitter
  black holes in modified gravity
  We investigate the future evolution of Nariai black hole which is extremal
limit of Schwarzschild-de Sitter one in modified gravity. The perturbations
equations around Nariai black hole are derived in static and cosmological
patches for general $F(R)$-gravity. The analytical and numerical study of
several realistic $F(R)$-models shows the occurence of rich variety of
scenarios: instabilities, celebrated Hawking evaporation and anti-evaporation
of black hole. The realization of specific scenario depends on the model under
consideration. It is remarkable that the presence of such primordial black
holes at current universe may indicate towards the modified gravity which
supports the anti-evaporation as preferrable model. As some generalization we
extend the study of Nariai black hole evolution to modified Gauss-Bonnet
gravity. The corresponding perturbations equations turn out to be much more
complicated than in the case of $F(R)$ gravity. For specific example of
modified Gauss-Bonnet gravity we demonstrate that Nariai solution maybe stable.
