  Analytical formulas for next-to-leading order temperature corrections to the
thermal state variables of interacting nucleons in bulk matter are derived in
the degenerate limit. The formalism developed is applicable to a wide class of
non-relativistic and relativistic models of hot and dense matter currently used
in nuclear physics and astrophysics (supernovae, proto-neutron stars and
neutron star mergers) as well as in condensed matter physics. We consider the
general case of arbitrary dimensionality of momentum space and an arbitrary
degree of relativity (for relativistic mean-field theoretical models). For
non-relativistic zero-range interactions, knowledge of the Landau effective
mass suffices to compute next-to-leading order effects, but in the case of
finite-range interactions, momentum derivatives of the Landau effective mass
function up to second order are required. Numerical computations are performed
to compare results from our analytical formulas with the exact results for
zero- and finite-range potential and relativistic mean-field theoretical
models. In all cases, inclusion of next-to-leading order temperature effects
substantially extends the ranges of partial degeneracy for which the analytical
treatment remains valid.
