  We point out that the lightest Kaluza-Klein particle (LKP) dark matter in
universal extra dimension (UED) models efficiently annihilates through the
coannihilation process including the first KK Higgs bosons when the Higgs mass
is slightly heavy as 200 - 230 GeV, which gives the large Higgs self-coupling.
The large self-coupling naturally leads the mass degeneracy between the LKP and
the first KK Higgs bosons and large annihilation cross sections of the KK Higgs
bosons. These are essential for the enhancement of the annihilation of the LKP
dark matter, which allows large compactification scale \sim 1 TeV to be
consistent with cosmological observations for the relic abundance of dark
matter. We found that the thermal relic abundance of the LKP dark matter could
be reconciled with the stringent constraint of electroweak precision
measurements in the minimal UED model.
