Let $X$ be a torus, $Z$ be one of the coordinate circles, and $U$ be the complement $X\setminus Z$. Consider the constant sheaf of $\mathbb{Q}$-valued functions on $U$, denoted $\mathbb{Q}_U$, and its extension by zero to the whole of $X$, $j_!\mathbb{Q}_U$. I'm interested in the cohomology of this sheaf.
We know we have a short exact sequence $0\to j_!\mathbb{Q}_U \to \mathbb{Q} \to i_*\mathbb{Q}_Z\to 0$, which (after some easy calculations) yields the long exact sequence $$0\to \mathbb{Q}\to\mathbb{Q}\to H^1(j_!\mathbb{Q}_U)\to \mathbb{Q}^{\oplus 2}\to \mathbb{Q}\to H^2(j_!\mathbb{Q}_U) \to \mathbb{Q}\to 0.$$
This is enough to tell us that $H^1(j_!\mathbb{Q}_U) \cong H^2(j_!\mathbb{Q}_U)$, but I'm not sure what else I can say. I tried using a Meyer-Vietoris argument with the sheaf $j_!\mathbb{Q}_U$, but wasn't able to calculate anything useful. What other tools would be useful here?
Note: It seems intuitive to me that there should be some difference in the cohomology depending on whether $Z$ is a coordinate circle as we assumed, or some simple loop on the torus. Where does this come into play?