  The cohomological rigidity problem for toric manifolds asks whether the
cohomology ring of a toric manifold determines the topological type of the
manifold. In this paper, we consider the problem with the class of one-twist
Bott manifolds to get an affirmative answer to the problem. We also generalize
the result to quasitoric manifolds. In doing so, we show that the twist number
of a Bott manifold is well-defined and is equal to the cohomological complexity
of the cohomology ring of the manifold. We also show that any cohomology Bott
manifold is homeomorphic to a Bott manifold. All these results are also
generalized to the case with $\mathbb Z_{(2)}$-coefficients, where $\mathbb
Z_{(2)}$ is the localized ring at 2.
