  A 125 GeV Higgs-like particle discovered at the LHC in 2012 has properties
expected for it in the Standard Model (SM), with a possible enhancement in the
two-photon channel. Such SM-like Higgs scenario can be realized within the
Inert Doublet Model (IDM) - a version of the Two Higgs Doublet Model with an
exact discrete D (Z_2-type) symmetry. In this model one SU(2) doublet plays the
role of the SM Higgs doublet with one SM-like Higgs boson. The second doublet
has no vacuum expectation value and does not interact with fermions. Among four
scalars constituting this D-odd doublet the lightest one is stable, being if
neutral a good DM candidate with the right relic density. In this paper an
analysis of the two-photon Higgs decay rate in IDM, respecting theoretical and
other experimental constraints, is presented. The enhancement in the two-photon
channel is possible only if invisible channels are closed, with the enhancement
R_{\gamma \gamma}>1.2 for masses of DM and charged scalars below 154 GeV. The
temperature evolution of the Universe to the Inert phase (described by the IDM)
in T^2 approximation is presented and all possible sequences of phase
transitions in one, two and three steps are analyzed. Going beyond this
approximation by using an effective potential approach with one-loop T=0
Weinberg-Coleman term and temperature dependent effective potential at T\not =0
we address the question, whether the strong first-order phase transition needed
for baryogenesis can be realized in IDM. A region with a good DM relic density
and a strong first-order phase transition is established, and discussed in the
light of the XENON-100 data.
