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").
I'd also like to point out that the group $H^0\big(G/B Regardless of characteristic, \mathcal L ( k[T] ) \big)$ has though, we have a simpler nice description using Frobenius reciprocityof the global sections. Indeed, for any simple $G$-module Let $L$, we have (denoting X^+(T)$denote the trivial module by set of dominant weights in$k$): X(T)$; then we get $$Hom_G \Big( L, H^0 \; H^0\big(G/B, big(G/B, \mathcal L ( k[T] ) \big) \Big) \cong Hom_G \Big( L, \; Ind_T^G k \Big) bigoplus_{\mu \cong Hom_Tin X^+(T)} H^0( LG/B, k ) \cong L_0 mathcal L(\mu) ) ,$$where $L_0$ denotes the $0$-weight space a direct sum of standard modules for $L$. Thus, regardless of the G$. In characteristic of k0 these modules are all simple, the multiplicity of any but they are not all simple module$L$in$ H^0\big(G/B, \mathcal L ( k[T] ) \big) $is precisely the dimension of the 0-weight space of$L$.positive characteristic. 1 Chris Brav's answer gives a nice description of the cohomology in the$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 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"). I'd also like to point out that the group$ H^0\big(G/B, \mathcal L ( k[T] ) \big) $has a simpler description using Frobenius reciprocity. Indeed, for any simple$G$-module$L$, we have (denoting the trivial module by$k$): $$Hom_G \Big( L, \; H^0\big(G/B, \mathcal L ( k[T] ) \big) \Big) \cong Hom_G \Big( L, \; Ind_T^G k \Big) \cong Hom_T( L, k ) \cong L_0 ,$$ where$L_0$denotes the$0$-weight space of$L$. Thus, regardless of the characteristic of k, the multiplicity of any simple module$L$in$ H^0\big(G/B, \mathcal L ( k[T] ) \big) $is precisely the dimension of the 0-weight space of$L\$.