Frobenius pushforwards of of vector bundles on projective spaces

We investigate when the filtration induced by Beilinson's spectral sequence splits non-canonically into a direct sum decomposition. We conclude that for any vector bundle $\mathcal{E}$ on a projective space over an algebraically closed field of characteristic $p>0$ there exists $r_{0}$ such that for $r\geq r_{0}$ the Frobenius pushforward $\mathsf{F}^{r}_{*}\mathcal{E}$ decomposes as a direct sum of line bundles and exterior powers of the cotangent bundle (we also give a variant for the "toric Frobenius map" valid in any characteristic). As an application we give a short proof of Klyachko's theorem for vanishing of the cohomology of toric vector bundles on projective spaces.