  We analyze the thermal history of the 2HDM and determine the parameter
regions featuring a first-order electroweak phase transition (FOEWPT) and also
much less studied phenomena like high-temperature electroweak (EW) symmetry
non-restoration and the possibility of vacuum trapping (i.e. the Universe
remains trapped in an EW-symmetric vacuum throughout the cosmological
evolution, despite at $T=0$ the EW breaking vacuum is deeper). We show that the
presence of vacuum trapping impedes a first-order EW phase transition in 2HDM
parameter-space regions previously considered suitable for the realization of
electroweak baryogenesis. Focusing then on the regions that do feature such a
first-order transition, we show that the 2HDM parameter space that would yield
a stochastic gravitational wave signal potentially detectable by the future
LISA observatory is very contrived, and will be well probed by direct searches
of 2HDM Higgs bosons at the HL-LHC, and (possibly) also via measurements of the
self-coupling of the Higgs boson at 125 GeV. This has an important impact on
the interplay between LISA and the LHC regarding the exploration of first-order
phase transition scenarios in the 2HDM: the absence of new physics indications
at the HL-LHC would severely limit the prospects of a detection by LISA.
Finally, we demonstrate that as a consequence of the predicted enhancement of
the self-coupling of the Higgs boson at 125 GeV the ILC would be able to probe
the majority of the 2HDM parameter space yielding a FOEWPT through measurements
of the self-coupling, with a large improvement in precision with respect to the
HL-LHC.
