  Extremal cosmological black holes are analysed in the framework of the most
general second order scalar-tensor theory, the~so-called Horndeski gravity.
Such~extremal black holes are a particular case of Schwarzschild-De Sitter
black holes that arises when the black hole horizon and the cosmological one
coincide. Such~metric is induced by a particular value of the effective
cosmological constant and is known as Nariai spacetime. The~existence of this
type of solutions is studied when considering the Horndeski Lagrangian and its
stability is analysed, where the so-called anti-evaporation regime is studied.
Contrary to other frameworks, the~radius of the horizon remains stable for some
cases of the Horndeski Lagrangian when considering perturbations at linear
order.
