  We exploit the many-body self-consistent Green's function method to analyze
finite-temperature properties of infinite nuclear matter and to explore the
behavior of the thermal index used to simulate thermal effects in equations of
state for astrophysical applications. We show how the thermal index is both
density and temperature dependent, unlike often considered, and we provide an
error estimate based on our ${\it ab~initio}$ calculations. The inclusion of
many-body forces is found to be critical for the density dependence of the
thermal index. We also compare our results to a parametrization in terms of the
density dependence of the nucleon effective mass. Our study questions the
validity of predictions made for the gravitational-wave signal from
neutron-star merger simulations with a constant thermal index.
