  Following previous study, in the Littlest Higgs model (LHM), the heavy photon
is supposed to be a possible dark matter candidate and its relic abundance of
the heavy photon is estimated in terms of the Boltzman-Lee-Weinberg
time-evolution equation. The effects of the T-parity violation is also
considered. Our calculations show that when Higgs mass $M_H$ taken to be 300
GeV and don't consider T-parity violation, only two narrow ranges
$133<M_{A_{H}}<135$ GeV and $167<M_{A_{H}}<169$ GeV are tolerable with the
current astrophysical observation and if $135<M_{A_{H}}<167$ GeV, there must at
least exist another species of heavy particle contributing to the cold dark
matter. As long as the T-parity can be violated, the heavy photon can decay
into regular standard model particles and would affect the dark matter
abundance in the universe, we discuss the constraint on the T-parity violation
parameter based on the present data. Direct detection prospects are also
discussed in some detail.
