sheaf theory - vanishing theorems

I would be glad to know about possible generalizations of the following results:

1) (Grothendieck) Let $X$ be a noetherian topological space of dimension $n$. Then for all $i>n$ and all sheaves of abelian groups $cal{F}$ on $X$, we have $H^i(X; cal{F})=$ 0. [See Hartshorne, Algebraic Geometry, III.2.7.]

2) Let $X$ be an $n$-dimensional $C^0$-manifold. Then for all $i>n$ and all sheaves of abelian groups $cal{F}$ on $X$, we have $H^i(X; cal{F})=$ 0 . [See Kashiwara-Schapira, Sheaves on manifolds, III.3.2.2]

More precisely, I'm interested in dropping the "abelian groups" hypothesis: could I take sheaves in any, say, AB5 abelian category?

Apparently, in Grothendieck's theorem, the "abelian groups" hypothesis is necessary -at least in Hartshorne's proof-, because at the end you see a big constant sheaf $mathbf{Z}$. But what happens if we talk about sheaves of $R$-modules, with $R$ any commutative ring with unit, for instance?

Are those generalizations trivial ones? False for trivial reasons?

Any hints or references will be welcome.