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][1]" 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”][1]* by Christian Houzel. (English) 
Grundlehren der Mathematischen Wissenschaften, 292. Berlin etc.: Springer-Verlag, pp. x+512 (1990), [MR1074006](https://mathscinet.ams.org/mathscinet-getitem?mr=1074006), [Zbl 0709.18001](https://zbmath.org/0709.18001).


  [1]: http://books.google.com/books?id=qfWcUSQRsX4C&lpg=PA475&ots=Dk-Z-JgqhP&dq=kashiwara%20schapira%20sheaves%20on%20manifolds&pg=PP1#v=onepage&q=&f=false