Stability and existence analysis of static black holes in pure Lovelock theories

In this paper, we study existence and stability of static black holes in Lovelock theories with a particular focus on pure Lovelock black holes. We derive the equation of stability from action without using S-deformation approach. It turns out that though pure-Lovelock black hole in even dimension is always unstable, however introduction of $\Lambda$ stablizes it by prescribing a lower threshold mass while there also exists an upper bound on mass which is given by existence of horizon. We also study stability of dimensionally continued black holes as well as of pure Lovelock analogue of BTZ black hole in all odd dimensions.