  By means of group theory, topological quantum chemistry, first-principles and
Monte Carlo calculations, we analyze the topology of the 2D buckled honeycomb
lattice phonon spectra. Taking the pure crystal structure as an input, we show
that eleven distinct phases are possible, five of which necessarily have
non-trivial topology according to topological quantum chemistry. Another four
of them are also identified as topological using Wilson loops in an analytical
model that includes all the symmetry allowed force constants up to third
nearest neighbors, making a total of nine topological phases. We then compute
the ab initio phonon spectra for the two-dimensional crystals of Si, Ge, P, As
and Sb in this structure and construct its phase diagram. Despite the large
proportion of topological phases found in the analytical model, all of the
crystals lie in a trivial phase. By analyzing the force constants space using
Monte Carlo calculations, we elucidate why topological phonon phases are
physically difficult to realize in real materials with this crystal structure.
