Chris Brav's answer gives a nice description of the cohomology in the $D$-module context. Just to expand a bit on that, I'd like to give a direct description, and also say a word about positive characteristic, since the paper of Levasseur-Stafford only discusses the characteristic-0 case.

As Jason Starr mentions above, we can decompose this cohomology in terms of line bundles on $G/B$. Fix an isomorphism $B/U \cong T$; in particular, this gives $T$ and $k[T]$ structures of $B$-modules. For any $B$-module $M$ let $ \mathcal L(M) $ denote the $G$-equivariant bundle on $G/B$ with fiber $M$. Then we have a $G$-equivariant isomorphism $$ H^*(G/U, \mathcal O_{G/U}) \cong H^* \big(G/B, \mathcal L ( k[T] ) \big) . $$ So, we basically want to understand the structure of $k[T]$ as a $B$-module.

Let $X(T)$ denote the character group of $T$. Then $k[T] \cong k( X(T) )$, the group algebra of $X(T)$ over $k$. As Jason pointed out, we now get a direct sum of line bundles on $G/B$ corresponding to the elements of $X(T)$. However, note that we may not get *all* of the line bundles on $G/B$, since $G$ might not be simply-connected; $X(T)$ may be a proper subset of the full weight lattice of $G$. Here isogeny will play a role. (In Levasseur-Stafford, for example, they assume $G$ to be simply connected). In any event, the characteristic 0 story will now follow from Borel-Weil. The postive-characteristic answer, on the other hand, is still an open question, since the full cohomology of line bundles on $G/B$ isn't completely known there (although a lot is known, cf Jantzen's book "Representations of Algebraic Groups").

Regardless of characteristic, though, we have a nice description of the global sections. Let $X^+(T)$ denote the set of dominant weights in $X(T)$; then we get $$ H^0 \big(G/B, \mathcal L ( k[T] ) \big) \cong \bigoplus_{\mu \in X^+(T)} H^0( G/B, \mathcal L(\mu) ) , $$ a direct sum of standard modules for $G$. In characteristic 0 these modules are all simple, but they are not all simple in positive characteristic.