  Over a field of prime characteristic $p>2$, we prove that the cohomology
rings of some pointed Hopf algebras of dimension $p^3$ are finitely generated.
These are Hopf algebras arising in the ongoing classification of finite
dimensional Hopf algebras in positive characteristic, and include bosonizations
of Nichols algebras of Jordan type in a general setting as well as their
liftings when $p=3$. Our techniques are applications of twisted tensor product
resolutions and Anick resolutions in combination with May spectral sequences.
