  We study the importance of the thermal behavior of the hadron-quark phase
transition in neutron star (NS) mergers. To this end, we devise a new scheme
approximating thermal effects to supplement any cold, barotropic hybrid
equation of state (EoS) model, i.e. two-phase EoS constructions with a hadronic
regime and a phase of deconfined quark matter. The consideration of
temperature-dependent phase boundaries turns out to be critical for a
quantitative description of quark matter effects in NS mergers, since the
coexistence phase can introduce a strong softening of the EoS at finite
temperature, which is even more significant than the change of the EoS by the
phase transition at T=0. We validate our approach by comparing to existing
fully temperature-dependent EoS models and find a very good quantitative
agreement of postmerger gravitational-wave (GW) features. Simulations with the
commonly-used thermal ideal-gas approach exhibit sizable differences compared
to full hybrid models implying that its use in NS merger simulations with quark
matter is problematic. Our new scheme provides the means to isolate thermal
effects of quark matter from the properties of the cold hybrid EoS and thus
allows an assessment of the thermal behavior alone. We show that different
shapes of the phase boundaries at finite temperature can have a large impact on
the postmerger dynamics and GW signal for the same cold hybrid model. This
finding demonstrates that postmerger GW emission contains important
complementary information compared to properties extracted from cold stars. We
also show by concrete examples that it is even possible for quark matter to
only occur and thus be detectable in finite-temperature systems like merger
remnants but not in cold NSs (abbreviated).
