The sheaf cohomology ${H}^i(X,F)$ of a (topological) manifold $X$ of dimension $n$ vanishes for $i > n$. This is a topological version of Grothendieck's vanishing theorem above. You can find this result in Kashiwara-Schapira's "Sheaves on manifolds" proposition III.3.2.2.
Reference
Masaki Kashiwara, Pierre Schapira, [Houzel, Christian] Sheaves on manifolds. With a short history “Les débuts de la théorie des faisceaux” by Christian Houzel. (English) Grundlehren der Mathematischen Wissenschaften, 292. Berlin etc.: Springer-Verlag, pp. x+512 (1990), MR1074006, Zbl 0709.18001.
