  We perform a comprehensive study of a model in which the Higgs sector is
extended to contain two Higgs doublet fields, with the four types of
possibilities to couple to standard fermions, as well as an additional light
pseudoscalar Higgs boson which mixes with the one of the two doublets. This
2HD+a model includes also a stable isosinglet massive fermion that has the
correct thermal relic abundance to account for the dark matter in the Universe.
We summarize the theoretical constraints to which the model is subject and then
perform a detailed study of the phenomenological constraints. In particular, we
discuss the bounds from the LHC in the search for light and heavy scalar
resonances and invisible states and those from high precision measurements in
the Higgs, electroweak and flavor sectors, addressing the possibility of
explaining the deviation from the standard expectation of the anomalous
magnetic moment of the muon and the $W$-boson mass recently observed at
Fermilab. We also summarize the astrophysical constraints from direct and
indirect detection dark matter experiments. We finally conduct a thorough
analysis of the cosmic phase transitions and the gravitational wave spectrum
that are implied by the model and identify the parameter space in which the
electroweak vacuum is reached after single and multiple phase transitions. We
then discuss the prospects for observing the signal of such gravitational waves
in near future experiments such as LISA, BBO or DECIGO.
