  We show that the relic abundance of the minority component of asymmetric dark
matter can be very sensitive to the expansion rate of the Universe and the
temperature of transition between a non-standard pre-Big Bang Nucleosynthesis
cosmological phase and the standard radiation dominated phase, if chemical
decoupling happens before this transition. In particular, because the
annihilation cross section of asymmetric dark matter is typically larger than
that of symmetric dark matter in the standard cosmology, the decrease in relic
density of the minority component in non-standard cosmologies with respect to
the majority component may be compensated by the increase in annihilation cross
section, so that the annihilation rate at present of asymmetric dark matter,
contrary to general belief, could be larger than that of symmetric dark matter
in the standard cosmology.
