  We study in detail the model by Isidori and Kamenik that is claimed to
explain the top quark forward-backward asymmetry at Tevatron, provide GeV-scale
dark matter (DM), and possibly improve the agreement between data and theory in
Tevatron W+jj events. We compute the DM thermal relic density, the
spin-independent DM-nucleon scattering cross section, and the cosmic microwave
background constraints on both Dirac and Majorana neutralino DM in the
parameter space that explains the top asymmetry. A stable light neutralino is
not allowed unless the local DM density is 3-4 times smaller than expected, in
which case Dirac DM with mass around 3 GeV may be possible, to be tested by the
Planck mission. The model predicts a too broad excess in the dijet distribution
and a strong modification of the missing E_T distribution in W+jj events.
