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It has been proved by Thomsen [Thomsen J. F.. “Frobenius direct images of line bundles on toric varieties” Journal of Algebra 226, no. 2 (2000)] that such push-forward is a direct sum of line bundles. To be precise, his theorem applies to the Frobenius map on a smooth toric variety in characteristic $p$, but the same result holds for the "toric Frobenius" map you describe. For an explicit description, see Section 2 of [Achinger P. "A Characterization of Toric Varieties in Characteristic p" Int Math Res Notices (2015)]. See also Bondal's article, p. 284 in Oberwolfach Reports from 2006: https://www.mfo.de/document/0605/OWR_2006_05.pdf .