Let $\mathcal F$ and $\mathcal G$ be sheaves on topological spaces $X$ and $Y$ respectively. The external tensor product of $\mathcal F$ and $\mathcal G$ is the sheaf on $X\times Y$ defined as $\mathcal F\boxtimes\mathcal G \cong \pi_1^*\mathcal F \otimes\pi_2^*\mathcal G $. Can we say anything about the cohomology $H^k (\mathcal F\boxtimes\mathcal G)$ in terms of the cohomology of $\mathcal F$ and $\mathcal G$? I've done some small, concrete examples but I'm interested in a general statement.
EDIT: Is there any hope at all if in particular $\mathcal F$ and $\mathcal G$ are not quasi-coherent? Say, for example, if they are extension-by-zero sheaves?
Denote by $f\colon X\times Y\to X,g\colon X\times Y\to Y$ the natural projections. Grothendick spectral sequence has $E_2=H^i(X,R^j f_*(f^*\mathcal F\otimes g^*\mathcal G))$. This spectral sequnce converges to $H^{i+j}(X\times Y,f^*\mathcal F\otimes g^*\mathcal G))$. Projection formula gives: $E_2=H^i(X,\mathcal F\otimes R^j f_*(g^*\mathcal G))$. Assume that $Y$ is compact. Then proper Base change (https://stacks.math.columbia.edu/tag/09V4) gives $E_2=H^i(X,\mathcal F\otimes H^j(Y,\mathcal G))$. Now if our sheaves are vector spaces over some field $k$ then $H^i(X,\mathcal F\otimes H^j(Y,\mathcal G))$ is non-canonically isomorphic to $H^i(X,\mathcal F)\otimes_k H^j(Y,\mathcal G)$. So we have
$$\dim_k(H^n(X\times Y,f^*\mathcal F\otimes g^*\mathcal G))\leq \sum\limits_{k=0}^nH^k(X,\mathcal F)\otimes_k H^{n-k}(Y,\mathcal G).$$