Critical points of $D$-dimensional extended gravities

We study the parameter space of $D$-dimensional cosmological Einstein gravity together with quadratic curvature terms. In $D>4$ there are in general two distinct (anti)-de Sitter vacua. We show that, for an appropriate choice of the parameters, there exists a critical point for one of the vacua, with only massless tensor, but neither massive tensor nor scalar, gravitons. At criticality, the linearized excitations have formally vanishing energy (as do black hole solutions). A further restriction of the parameters gives a one-parameter cosmological Einstein plus ${\mathrm{\text{Weyl}}}^2$ model with a unique vacuum, whose $\Lambda$ is determined.