  We study selected aspects of quantum gravity phenomenology inspired by the
gravitational analogy in Bose--Einstein condensates (BECs). We first review the
basic ideas and formalism of analogue gravity in BECs, with particular emphasis
on the possibility of simulating black holes. The non-relativistic,
'superluminal' modifications of the dispersion relation in a BEC beyond the
hydrodynamic limit make it a particularly interesting model for many scenarios
of quantum gravity phenomenology which consider a possible violation of local
Lorentz invariance at high energies. In particular, these modifications allow
the study of kinematical corrections that such quantum gravity scenarios could
impose on general relativity.
  A simple (1+1)-dimensional acoustic black hole configuration in a BEC is
presented, and its dynamical stability and quasinormal mode spectrum are
studied. Then, an analysis is performed of the Hawking radiation for a
collapsing geometry in which a black hole is created. It is seen that the
superluminality of the dispersion relation leads to a frequency-dependence of
the horizon, which can cause strong qualitative and quantitative modifications
with respect to the standard (Lorentz-invariant) Hawking spectrum. We end with
some considerations related to the possibility of constructing a serious toy
model for Planck-scale gravity understood as an emergent phenomenon, based on
the condensed matter analogy. In particular, we discuss the problem of
diffeomorphism invariance in such a seemingly background-dependent approach and
indicate some possible ideas for how to recover the Einstein equations in the
adequate limit.
