I would like to know if the odd Betti numbers of a projective bundle P(E) for some vector bundle E over say a compact complex smooth algebraic variety B are zero just as in the case for ordinary projective spaces over Spec(k), or more generally how to generalize standard calculations of the cohomology of projective space to projective bundles.

If $E$ is of rank $r$ then $H^i(P_B(E)) = \sum_{t = 0}^{r1} H^{i2t}(B)$ (where the summands with negative $i  2t$ are omitted). So $H^{odd}(P_B(E)) = 0$ if and only if $H^{odd}(B) = 0$. 

