algebraic groups - Duflot-type theorem for Hopf algebras ?

In group cohomology Duflot's theorem states that the depth of the mod p cohomology ring of a finite group is greater than or equal to the p-rank of the center of a Sylow p-subgroup.

Is there a corresponding result for the cohomology of a finite dimensional cocommutative Hopf algebra (or, equivalently, a finite group scheme) ?

Any hint is appreciated.