  Observations of isolated neutron stars place constraints on the equation of
state (EOS) of cold, neutron-rich matter, while nuclear physics experiments
probe the EOS of hot, symmetric matter. Many dynamical phenomena, such as
core-collapse supernovae, the formation and cooling of proto-neutron stars, and
neutron star mergers, lie between these two regimes and depend on the EOS at
finite temperatures for matter with varying proton fractions. In this paper, we
introduce a new framework to accurately calculate the thermal pressure of
neutron-proton-electron matter at arbitrary density, temperature, and proton
fraction. This framework can be expressed using a set of five
physically-motivated parameters that span a narrow range of values for
realistic EOS and are able to capture the leading-order effects of degenerate
matter on the thermal pressure. We base two of these parameters on a new
approximation of the Dirac effective mass, with which we reproduce the thermal
pressure to within <~30% for a variety of realistic EOS at densities of
interest. Three additional parameters, based on the behavior of the symmetry
energy near the nuclear saturation density, allow for the extrapolation of any
cold EOS in beta-equilibrium to arbitrary proton fractions. Our model thus
allows a user to extend any cold nucleonic EOS, including piecewise-polytropes,
to arbitrary temperature and proton fraction, for use in calculations and
numerical simulations of astrophysical phenomena. We find that our formalism is
able to reproduce realistic finite-temperature EOS with errors of <~20% and
offers a 1-3 orders-of-magnitude improvement over existing ideal-fluid models.
