EDIT. Corrected the statement ($\sigma$ should be $p-1$ times what I wrote) and answered the question in the comment.
In general, the push-forward of a line bundle on the flag variety $G/B$ will not be the direct sum of line bundles. (This holds only on toric varieties.)
However, Haboush [1] (and independently Andersen) has proved that this is true for some special values of $\lambda$. More specifically, he proved that for $\lambda = \sigma + p\mu$ where $\sigma$ is $(p-1)/2$ times the sum of the positive roots (which is $(p-1)$ times the sum of the fundamental weights), one has an equivariant isomorphism $$ F_* \mathcal{O}(\lambda) \simeq V\otimes \mathcal{O}(\mu) $$ where $V$ is the Steinberg module, irreducible of highest weight $\sigma$. In particular, $F_*\mathcal{O}(\sigma)$ is a trivial vector bundle $V\otimes \mathcal{O}_{G/B}$ corresponding to the Steinberg representation.
Notice that the statement follows from the case $\mu=0$ by the projection formula and the fact that $F^* L = L^p$ for a line bundle $L$: $$ F_* \mathcal{O}(\sigma + p\mu) = F_* (\mathcal{O}(\sigma)\otimes F^* \mathcal{O}(\mu)) = (F_* \mathcal{O}(\sigma))\otimes \mathcal{O}(\mu) = V\otimes \mathcal{O}(\mu). $$
Now, to answer the question in the comment: in the reference, they are interested in $F_* \mathcal{O}(-\rho)$ where $\rho$ is the sum of the fundamental weights. Since $\sigma = (p-1)\rho = -\rho + p\rho$, we obtain with $\lambda = -\rho = \mu$: $$ F_* \mathcal{O}(-\rho) \simeq V\otimes \mathcal{O}(-\rho). $$
This isomorphism (saying that $\mathcal{O}(-\rho)$ is an "eigenvector" for $F_*$ with "eigenvalue" $V$) is employed in Samokhin's paper [2], which you may find useful.
[1] W. J. Haboush, A short proof of the Kempf vanishing theorem, Inventiones mathematicae volume 56, pages 109–112 (1980)