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Let $p$ and $q$ be distinct prime numbers. We study the Galois objects andcocycle deformations of the noncommutative, noncocommutative, semisimple Hopfalgebras 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 hisprevious results for the 8-dimensional Kac-Paljutkin Hopf algebra. This is doneas a consequence of the classification of categorical Morita equivalenceclasses among semisimple Hopf algebras of odd dimension $p^3$, established bythe third-named author in an appendix.

Cocycle deformations and Galois objects for semisimple Hopf algebras of dimension $p^3$ and $pq^2$