Let $C$ be a curve of genus $g \geq 1$ and let $J^d$ be its degree $d$ Jacobian.

Inside of $J^{g-1}$ there is the Theta divisor $\Theta$, which can be defined in various ways; the quickest definition is probably: it's the image of the Abel-Jacobi map $C^{(g-1)} \to J^{g-1}$ sending an effective degree $g-1$ divisor to the corresponding line bundle. Picking an isomorphism $J^{g-1} \cong J^d$, we also write $\Theta$ for the corresponding divisor in $J^d$.

> How to compute $H^\ast(J;\Theta)$, or $h^\ast(J;\Theta)$? Or alternatively, what is known about these groups?

I suspect this is something embarrassingly standard and/or obvious and/or well-known and/or classical, but I haven't been able to figure anything out. The only thing along these lines that I was able to figure out was how to compute the Euler characteristic $\chi(J;\Theta^k)$ where $k$ is an integer: By Hirzebruch-Riemann-Roch and the Poincare formula it's $$\int_J \operatorname{ch}(\Theta^k) = \int_J e^{k\theta} = \int_J k^g \theta^g / g! = k^g.$$