  Binary neutron star mergers probe the dense-matter equation of state (EoS)
across a wide range of densities and temperatures, from the cold conditions of
the inspiral to the high-temperature matter of the massive neutron star
remnant. In this paper, we explore the sensitivity of neutron star mergers to
uncertainties in the finite-temperature part of the EoS with a series of merger
simulations performed in full general relativity. We expand on our previous
work to explore the interplay between the thermal prescription and the
stiffness of the zero-temperature EoS, which determines the compactness of the
initial neutron stars. Using a phenomenological model of the particle effective
mass, $M^*$, to calculate the finite-temperature part of the EoS, we perform
merger simulations for a range of thermal prescriptions, together with two cold
EoSs that predict either compact or large-radius initial neutron stars. We
report on how the choice of $M^*$-parameters influences the thermal properties
of the post-merger remnant, and how this varies for stars with different
initial stellar compactness. We characterize the post-merger gravitational wave
signals, and find differences in the peak frequencies of up to 190 Hz depending
on the choice of $M^*$-parameters. Finally, we find that the total dynamical
ejecta is in general only weakly sensitive to the thermal prescription, but
that a particular combination of $M^*$-parameters, together with a soft cold
EoS, can lead to significant enhancements in the ejecta.
