  In this paper we study the geometry of toric degeneration of a
Bott-Samelson-Demazure-Hansen (BSDH) variety. We give some applications to BSDH
varieties. Precisely, we classify Fano, weak Fano and log Fano BSDH varieties
and their toric limits in Kac-Moody setting. We prove some vanishing theorems
for the cohomology of tangent bundle (and line bundles) on BSDH varieties. We
also recover the results in arXiv:1604.01998, by toric methods.
