  We discuss dark matter (DM) physics in the Type-I inert two-Higgs-doublet
model (2HDM) with local U(1)_H Higgs gauge symmetry. The local U(1)_H gauge
symmetry is assigned to the extra Higgs doublet in order to avoid the
Higgs-mediated flavor problems, and it spontaneously breaks down to its
discrete subgroup. The lightest neutral scalar component H of the
U(1)_H-charged Higgs doublet, which does not have Yukawa couplings with the
Standard-Model (SM) fermions, is stable because of the remnant discrete
symmetry, and it interacts with the SM particles through the U(1)_H gauge boson
(Z_H) exchange as well as the SM boson exchange. We first investigate the
constraint on the U(1)_H gauge interaction, especially through the kinetic and
mass mixing between the SM gauge bosons and the extra gauge boson. Then we
discuss dark matter physics in our 2HDM: thermal relic density, and
direct/indirect detections of dark matter. The additional U(1)_H gauge
interaction plays a crucial role in reducing the DM thermal relic density. The
most important result within the inert DM model with local U(1)_H symmetry is
that ~ O(10) GeV dark matter scenario, which is strongly disfavored in the
usual Inert Doublet Model (IDM) with Z_2 symmetry, is revived in our model
because of newly open channels, H H -> Z_H Z_H , Z_H Z. Exotic Higgs decays, h
-> Z_H Z_H, Z Z_H, would be distinctive signatures of the inert 2HDM with local
U(1)_H symmetry.
