Is sheaf cohomology an invariant of the weak homotopy type? More precisely let $R$ be a commutative ring and $f:X\rightarrow Y$ a weak homotopy equivalence. Does it follow, that the induced maps $H^n(Y,\underline R) \rightarrow H^n(X,\underline R)$ are isomorphisms?

Edit: Since any space is weakly homotopy equivalent to a CW-complex, CW-complexes are locally contractible and for locally contractible spaces sheaf and singular cohomology coincide, a positive answer to this question would imply that sheaf cohomology and singular cohomology coincide for any space. This seems unlikely, but I don't know a counter example.