  Building on the work of Fyodorov (2004) and Fyodorov and Nadal (2012) we
examine the critical behaviour of population of saddles with fixed instability
index $k$ in high dimensional random energy landscapes. Such landscapes consist
of a parabolic confining potential and a random part in $N\gg 1$ dimensions.
When the relative strength $m$ of the parabolic part is decreasing below a
critical value $m_c$, the random energy landscapes exhibit a glass-like
transition from a simple phase with very few critical points to a complex phase
with the energy surface having exponentially many critical points. We obtain
the annealed probability distribution of the instability index $k$ by working
out the mean size of the population of saddles with index $k$ relative to the
mean size of the entire population of critical points and observe toppling of
stability hierarchy which accompanies the underlying glass-like transition. In
the transition region $m=m_c + \delta N^{-1/2}$ the typical instability index
scales as $k = \kappa N^{1/4}$ and the toppling mechanism affects whole
instability index distribution, in particular the most probable value of
$\kappa$ changes from $\kappa = 0$ in the simple phase ($\delta > 0 $) to a
non-zero value $ \kappa_{\max} \propto (-\delta)^{3/2}$ in the complex phase
($\delta < 0$). We also show that a similar phenomenon is observed in random
landscapes with an additional fixed energy constraint and in the $p$-spin
spherical model.
