In this article we prove that if $G$ is a connected, simply connected, semisimple algebraic group over an algebraically closed field of sufficiently large characteristic, then all the blocks of the restricted enveloping algebra $(U\mathfrak{g}{)}_0$ of the Lie algebra $\mathfrak{g}$ of $G$ can be endowed with a Koszul grading (extending results of Andersen, Jantzen, and Soergel). We also give information about the Koszul dual rings. In the case of the block associated to a regular character $\lambda$ of the Harish-Chandra center, the dual ring is related to modules over the specialized algebra $(U\mathfrak{g}{)}^{\lambda }$ with generalized trivial Frobenius character. Our main tool is the localization theory developed by Bezrukavnikov, Mirković, and Rumynin