  In this paper we provide the criteria for any generally covariant, parity
preserving, and torsion free theory of gravity to possess a stable de Sitter
(dS) or anti-de Sitter (AdS) background. By stability we mean the absence of
tachyonic or ghost-like states in the perturbative spectrum that can lead to
classical instabilities and violation of quantum unitarity. While we find that
the usual suspects, the F(R) and F(G) theories, can indeed possess consistent
(A)dS backgrounds, G being the Gauss-Bonnet term, another interesting class of
theories, string-inspired infinite derivative gravity, can also be consistent
around such curved vacuum solutions. Our study should not only be relevant for
quantum gravity and early universe cosmology involving ultraviolet physics, but
also for modifications of gravity in the infra-red sector vying to replace dark
energy .
