Let R denote the real line with its usual topology. Does there exist a sheaf F of abelian groups on R whose second cohomology group H^2(R,F) is non-zero? What about H^j(R,F) for integers j>=2 ?
(Here cohomology means derived functor cohomology as in, say, Hartshorne or EGA. Anyway this cohomology coincides with Cech cohomology since R is paracompact.)