  Extensions of the Higgs sector of the Standard Model allow for a rich
cosmological history around the electroweak scale. We show that besides the
possibility of strong first-order phase transitions, which have been thoroughly
studied in the literature, also other important phenomena can occur, like the
non-restoration of the electroweak symmetry or the existence of vacua in which
the Universe becomes trapped, preventing a transition to the electroweak
minimum. Focusing on the next-to-minimal two-Higgs-doublet model (N2HDM) of
type II and taking into account the existing theoretical and experimental
constraints, we identify the scenarios of electroweak symmetry non-restoration,
vacuum trapping and first-order phase transition in the thermal history of the
Universe. We analyze these phenomena and in particular their relation to each
other, and discuss their connection to the predicted phenomenology of the N2HDM
at the LHC. Our analysis demonstrates that the presence of a global electroweak
minimum of the scalar potential at zero temperature does not guarantee that the
corresponding N2HDM parameter space will be physically viable: the existence of
a critical temperature at which the electroweak phase becomes the deepest
minimum is not sufficient for a transition to take place, necessitating an
analysis of the tunnelling probability to the electroweak minimum for a
reliable prediction of the thermal history of the Universe.
