Can a neutron star become a black hole via cooling? How much does thermal expansion affect neutron stars? Would the loss of temperature cause a neutron star to be more densely packed and thus collapse into a black hole?
 A: No (or at least not much). One of the essential properties of stars that are largely supported by degeneracy pressure, is that this pressure is independent of temperature and that is because although a neutron star may be hot, it has such a small heat capacity, it contains very little thermal energy$^{*}$.
When a neutron star forms, it cools extremely rapidly by the emission of neutrinos, on timescales of seconds. During this phase, the neutron star does contract a little bit (tens of per cent), but by the time its interior has cooled to a billion Kelvin, the interior neutrons are degenerate and the contraction is basically halted. It is possible that a (massive) neutron star could make the transition to a black hole before this point.
If it does not do so, then from there, the neutron star continues to cool (but actually possesses very little thermal energy, despite its high temperature), but this makes almost no difference to its radius.
$^{*}$ In a highly degenerate gas the occupation index of quantum states is unity up the Fermi energy and zero beyond this. In this idealised case, the heat capacity would be zero - no kinetic energy can be extracted from the fermions, since there are no free lower energy states. In practice, and at finite temperatures, there are fermions $\sim kT$ above the Fermi energy that can fall into the few free states at $\sim kT$ below the Fermi energy. However, the fraction of fermions able to do so is only $\sim kT/E_F$, where $E_F$ is the kinetic energy of fermions at the Fermi energy. At typical neutron star densities, this fraction is of order $T/10^{12}\ {\rm K}$, so is very small once neutron stars cool (within seconds) below $10^{10}$ K.
What this means is that the heat capacity is extremely small and that whilst the neutrons in a neutron star contain an enormous reservoir of kinetic energy (thus providing a pressure), almost none of this can be extracted as heat.
A: The answer by @RobJeffries is correct, at least 95-99% of the time (and possibly all the time), and also the best answer to this question.  But just for the curious, some people do talk about newly formed, meta-stable neutron stars (usually in the form of 'magnetars') which collapse afte short periods of time from the combination of cooling and rotational deceleration.  Most models suggest that the rotational deceleration is a more important factor which leads to the collapse: the newly formed neutron stars can be spinning near 'breakup velocity'---where material on the equator is rotating quickly enough that it almost detaches.  The strong magnetic fields of magnetars, however, are believed to be effective at transferring angular momentum out, and slowing down this rotation.  The material on the NS loses that rotational support, and thus NS which were on the verge of collapse may cross the threshold and turn into a black-hole.
This model, which is highly theoretical, is used to explain extended emission in some Gamma Ray Bursts (which, people think, require black holes to be formed).  The idea is a meta-stable magnetar is formed, which continues to power xray (and some other) emission and ejecta, before later collapsing to a black-hole (after 10s to 100s of seconds, usually).
Lü et al. 2015 - The Millisecond Magnetar Central Engine in short GRBs 
Rowlinson et al. 2013 - Signatures of magnetar central engines in short GRB light curves 
Lasky et al. 2013 - Nuclear Equation of State from Observations of Short Gamma-Ray Burst Remnants
