  In this article, we systematically investigate the stability properties of
certain warped product Einstein manifolds. We characterize stability of these
metrics in terms of an eigenvalue condition of the Einstein operator on the
base manifold. In particular, we prove that all complete manifolds carrying
imaginary Killing spinors are strictly stable. Moreover, we show that
Ricci-flat and hyperbolic cones over K\"ahler-Einstein Fano manifolds and over
nonnegatively curved Einstein manifolds are stable if the cone has dimension
$n\geq10$.
