  Let $p$ and $q$ be distinct prime numbers. We study the Galois objects and
cocycle deformations of the noncommutative, noncocommutative, semisimple Hopf
algebras of odd dimension $p^3$ and of dimension $pq^2$. We obtain that the
$p+1$ non-isomorphic self-dual semisimple Hopf algebras of dimension $p^3$
classified by Masuoka have no non-trivial cocycle deformations, extending his
previous results for the 8-dimensional Kac-Paljutkin Hopf algebra. This is done
as a consequence of the classification of categorical Morita equivalence
classes among semisimple Hopf algebras of odd dimension $p^3$, established by
the third-named author in an appendix.
