Consider a field $K$ (of characteristic 0, say) and its absolute galois group $G_K^{ab} = Gal(\overline{K}/K)$, given the Krull topology: $U_E(\sigma) = \sigma Gal(\overline{K}/E)$ form a basis of the topology, ranging over $\sigma \in G_K^{ab}$ and $E/K$ finite galois.

Fix a group $G$ and denote by $R_E$ its representation ring over $E$, and by $R_E^\sigma \subset R_E$ the elements of $R_E$ fixed by $\sigma$.

We can construct a sheaf $\mathcal{F}$ on $G_K^{ab}$ by setting $\mathcal{F}(U_E(\sigma)) = R_E^\sigma$. It is a simple exercise to verify the axioms.

One might hope that the sheaf cohomology of $\mathcal{F}$ encodes information about the splitting behaviour of representations of G over various ground fields, but this is not the case: $G_K^{ab}$ is known to be totally disconnected, hausdorff and compact. It is a theorem [1, 5.1] that $H^r(G_K^{ab}, \mathcal{F}) = 0$ for $r > 0$. Furthermore the $U_E(\sigma)$ are actually clopen, so most useful subsets I can think of are also compact, hence their cohomology is equally uninteresting.

Is there a way to produce a useful cohomology along these lines?

Here "useful" essentially means "non-trivial", and "along these lines" basically "involving the galois action on $R_E$ for various $E$".