Is there a formula for $H^i(X,\mathcal{F}\otimes\mathcal{G})$? That is, a formula in terms of the $H^i(X,\mathcal{F})$ and $H^i(X,\mathcal{G})$? In my setting $X$ is a smooth, projective variety, but any information on, say, a topological setting will be helpful.
Disclaimer: a couple of similar questions address the specific question of vanishing or the issue of the Kunneth formula for $X\times X$ and $\mathcal{F}\boxtimes\mathcal{G}$, but those are not the same question as mine.