Let $X$ be a scheme (we can assume $X$ is smooth over a field $k$). Let $\mathcal F_1$ and $\mathcal F_2$ be two sheaves of abelian groups on $X$.
Is it always true that $H^i_{\text{flat}}(X, \mathcal F_1 \oplus \mathcal F_2) = H^i_{\text{flat}}(X, \mathcal F_1) \oplus H^i_{\text{flat}}(X, \mathcal F_2)$? If not in general, is it true in particular that $H^i_{\text{flat}}(X, \mu_{n} \times \mu_n) = H^i_{flat}(X, \mu_n ) \oplus H^i_{\text{flat}}(X, \mu_n )$, or if we assume extra conditions on $X$ (like quasi-compactness etc. …)?