  Binary neutron star mergers provide a unique probe of the dense-matter
equation of state (EoS) across a wide range of parameter space, from the
zero-temperature EoS during the inspiral to the high-temperature EoS following
the merger. In this paper, we implement a new model for calculating
parametrized finite-temperature EoS effects into numerical relativity
simulations. This "M* model" is based on a two-parameter approximation of the
particle effective mass and includes the leading-order effects of degeneracy in
the thermal pressure and energy. We test our numerical implementation by
performing evolutions of rotating single stars with zero- and non-zero
temperature gradients, as well as evolutions of binary neutron star mergers. We
find that our new finite-temperature EoS implementation can support stable
stars over many dynamical timescales. We also perform a first parameter study
to explore the role of the M* parameters in binary neutron star merger
simulations. All simulations start from identical initial data with identical
cold EoSs, and differ only in the thermal part of the EoS. We find that both
the thermal profile of the remnant and the post-merger gravitational wave
signal depend on the choice of M* parameters, but that the total merger ejecta
depends only weakly on the finite-temperature part of the EoS across a wide
range of parameters. Our simulations provide a first step toward understanding
how the finite-temperature properties of dense matter may affect future
observations of binary neutron star mergers.
