  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.
