Compute $H^i(S,\underline{\text{Hom}}(A,\mathbb G_m))$ for a semi-abelian scheme $A$

Question

How can I compute $H^i(S,\underline{\text{Hom}}(A,\mathbb G_m))$ (where $A$ denotes a semi-abelian scheme over $S$, $\mathbb G_m$ denotes the multiplicative group over $S$ and $\underline{\text{Hom}}$ denotes the Hom sheaf)?

The base scheme can be assumed to be noetherian. Even the case of a dvr would be fine.