I have been pondering on this question for half a year and haven't gotten an ideal answer. Hope someone can help!
Conjecture: Given a topological space $X$ and any dimension $i\in \mathbb{N}$, the Čech Cohomology (on the sheaf of maps) $\check{H}^i(X;\mathcal{F}_E)=0$.
Here $E$ is assumed to be a contractible topological group. The sheaf of maps on $E$ is defined as
$\forall U\subseteq_{\text{open}} X$, the sheaf is defined as $\mathcal{F}_E(U):=\text{Maps}(U,E)=\{f:U\to E\ |\ f\text{ is continuous} \}$
with a group structure $\forall f,g\in \mathcal{F}_E(U), (f\cdot g)(x):=f(x)\cdot g(x)$.
The restriction is defined as $\text{res}_{V,U}: \mathcal{F}_E(U) \to \mathcal{F}_E(V); f\mapsto f\circ i;$ for all $U,V\subseteq_{\text{open}} X$ such that $V\subseteq U$.
($i: V\to U$ is the inclusion.)