  We investigate the relic abundance of asymmetric dark matter in the
non-standard cosmological scenarios when the annihilation cross section
includes self annihilations. Here we discuss the kination model and brane world
cosmology. When the self annihilation is permitted for asymmetric dark matter,
there is possibility of washing out the pre-existed asymmetry. We find the
constraints on the cross section to avoid the complete washing out of the
asymmetry in the non-standard cosmological scenarios. The enhanced cosmic
expansion rate causes the freeze out point of wash-out to be earlier. The
larger self annihilation cross sections are allowed to exist in kination model
and brane world cosmology. Then, in the case of left-handed sneutrino
asymmetric dark matter, we find the value of the lower bound on winos mass is
smaller than that in the standard cosmological scenario.
