My understanding of a white dwarf star is that, although it hasn't happened yet due to the age of the universe, eventually as it loses energy due to radiating heat and tidal forces, it will cool to become a black dwarf - essentially a black ball of carbon.
What happens when a neutron star eventually cools? What is left, and do we know what it would look like?
The cooling history of a neutron star can be divided into an extremely rapid neutrino cooling phase, followed by an indefinitely long cooling phase due to the emission of photons from its surface.
The key to understanding neutron star cooling is to realise that degenerate neutrons possess almost no thermal energy, even when extremely hot - because the Pauli Exclusion principle forbids the neutrons from losing kinetic energy and falling into an already-occupied quantum state. Any cooling processes therefore act to reduce the temperature of the neutron star very quickly. Secondly, the interior of a neutron star is almost isothermal, due to the high thermal conductivity of degenerate gases, but the temperature at the surface is smaller by about a factor of 100 or so.
When a neutron star first forms, its internal temperature exceeds 10 billion kelvin, its surface temperature would be 100 million degrees and emit hard X-rays. Nevertheless, the small size of the neutron star limits radiative losses and instead it is neutrinos escaping from the interior that can cool it by a factor of around 100 in a few thousand years.
The neutrino losses proceed by something called the modified URCA process (somewhat faster direct URCA or processes involving pions/kaons maybe important in the first seconds/minutes of more massive neutron stars, but are blocked by complete neutron degeneracy and the comparative paucity of protons in the interior): cycles of neutron beta decay and inverse beta decay on protons produce anti-neutrinos and neutrinos. Bystander particles are required to conserve momentum. The temperature dependence of this process is $\sim T^{8}$, so if the temperature falls by a factor 100, the neutrino cooling rate falls by a factor of $10^{16}$.
Meanwhile, photon cooling from the surface, whilst initially negligible compared to neutrino cooling, falls only as $T^{4}$. After around 10,000-100,000 years, when the neutron star surface has cooled to about a million degrees, photon cooling, through soft X-ray emission dominates.
If nothing else happened to an isolated neutron star, it would continue to cool, such that $T \propto t^{-1/2}$. The vast majority of the estimated 1 billion neutron stars in our Galaxy, are in this state, whereas most of the pulsars we can observe are young and in the neutrino-cooling phase.
How cold can they get? The formula above suggests their temperature halves when their age quadruples. Following this, they could cool to the temperature of the Sun in around a billion years. However, there are reheating processes that can occur.
Isolated neutron stars can accrete material from the interstellar medium. This may be very little, since most neutron stars move fast. The gravitational potential energy is partly released as heat when it impacts the neutron star surface. Then, the low heat capacity of neutrons works in the opposite direction and comparatively little accretion is required to balance the photon losses.
Secondly, neutron stars are born as fast rotators and spin down. Angular momentum must be transferred outwards from the fluid interior to the outer crust. This is not a frictionless process. Thus some of the rotational kinetic energy can keep the star warm.
Thirdly, the neutron star has a huge magnetic field that gradually dissipates. Some of that energy may end up as heat in the neutron star through simple Ohmic dissipation of currents.
The bottom line is that the thermal emission from neutron stars older than about 100,000 years has not been observed, so we simply don't know how cool they might get. Some more information and references can be found in the Physics SE answer where I discuss where old neutron stars might appear on the HR diagram.
To begin with, it's important to discuss just how a neutron star cools. There are several related mechanisms involved in the main stage of a neutron star's thermal evolution, and they mainly rely on neutrino emission (primary reference Lim et al. (2015)).
First and foremost is the family of Urca processes. The direct Urca process (Durca) is $$\text{baryon }1\to\text{baryon }2+\text{lepton}+\text{antineutrino}$$ $$\text{baryon }2+\text{lepton}\to\text{baryon }1+\text{neutrino}$$ The baryons here are typically neutrons and protons, and the lepton is an electron, with its corresponding antineutrino. However, it can also happen with pions, kaons, and quark matter.
Next up is bremsstrahlung. This version is unlike the more commonly known variant where radiation is emitted from electrons. Instead, it is nucleon-nucleon bremsstrahlung, when two neutrons, two protons, or a proton and a neutron interact to produce a neutrino/antineutrino pair. In fact, one of the main differences between this bremsstrahlung and the Urca process is the lack of electrons in the former reaction.
Cooper pair decay takes place in neutron or proton pairs. The formation of the pairs is made possible by the extreme conditions inside the neutron star, which cause superfluidity and influence many of the other processes.
These three mechanisms take place in the core of the neutron star and throughout it. However, some other pathways take place only in the crust. They include
Now that we know the processes, what are the results?
The above processes are actually only dominant in the second of three stages in a neutron star's life (see Yakovlev et al.). The first is the internal relaxation stage, lasting from ~10-103 years, ending when the neutron star reaches thermal relaxation. The second is this neutrino cooling stage, lasting to ~105 years. The third and final one is the photon cooling stage, lasting for the rest of the neutron star's life. At this point, the star is simply getting cooler and cooler. It is not known how long this stage can last, but it doesn't appear that any boundary has been placed on it.
