  In all odd dimensions $\geq 5$ we produce examples of manifolds admitting
pairs of Sasaki structures with different basic Hodge numbers. In dimension $5$
we prove more precise results, for example we show that on connected sums of
copies of $S^2\times S^3$ the number of Sasaki structures with different basic
Hodge numbers within a fixed homotopy class of almost contact structures is
unbounded. All the Sasaki structures we consider are negative in the sense that
the basic first Chern class is represented by a negative definite form of type
$(1,1)$. We also discuss the relation of these results to contact topology.
