  The oscillation spectrum of a neutron star is notably rich and intrinsically
dependent on the equation of state of nuclear matter. With recent advancements
in gravitational-wave and electromagnetic astronomy, we are nearing the
capability to perform neutron-star asteroseismology and probe the complex
physics of neutron stars. With this in mind, we explore the implementation of
three-parameter finite-temperature matter models in the computation of
neutron-star oscillations. We consider in detail the thermodynamics of nuclear
matter and show how this information enters the problem. Our realistic
treatment takes into account entropy and composition gradients that exist in
the nuclear matter, giving rise to buoyant g-mode oscillations. To illustrate
the implementation, we determine the oscillation spectrum of a low-temperature
neutron star. In addition to the expected compositional and thermal g-modes, we
find perturbations sourced by phase transitions in the equation of state. We
also examine the thermal evolution of the oscillation spectrum of a neutron
star with constant redshifted temperature.
