  We compute the $F$-pure threshold of the natural cone over flag varieties in
characteristic $p>0$. Our calculations are mainly focused on flag varieties
that are arithmetically Gorenstein, but we offer some results in the
non-Gorenstein case. Our goal is to determine the $a$-invariant of the cone. As
a result, the $F$-pure thresholds we find are independent of the characteristic
$p$, hence one immediately gets the value of the log canonical threshold of
flags in characteristic 0 as well.
