  We present several deformation and rigidity results within the classes of
closed Riemannian manifolds which either are $2k$-Einstein (in the sense that
their $2k$-Ricci tensor is constant) or have constant $2k$-Gauss-Bonnet
curvature. The results hold for a family of manifolds containing all non-flat
space forms and the main ingredients in the proofs are explicit formulae for
the linearizations of the above invariants obtained by means of the formalism
of double forms.
