Watch that youtube link in my sibling comment first, but I actually wanted to take a crack at an answer. What's the change in mass, ∆m, as the star cools? We'll assume the mass of the neutron star is 1.5 solar masses.

    E = mc^2
    ∆m = ∆E/c^2

So really, what's ∆E? A hyper hand-wavy estimate:

    ∆E = Q = mc∆T (c being specific heat)

Specific heat by mass is really hard to predict, but by moles it is fairly constant, well within an order of magnitude. So we'll discuss mass in moles.

    m = neutron star moles
    m = (mass of neutron star / mass of neutron) / Avogadro's #
    m = 2.9580163e33 mol

    c = 24 J / (mol * K)

    ∆T = 1e12 K - 1e6 K

→

    ∆E = 24 * 2.9580163e33 * (1e12 - 1e6) J = 7.0991681e45 J

Substituting that back into the original, as a neutron star cools:

    ∆m = 7.0991681e45 J / c^2 = 7.89888982e28 kg

Which is like 2.6% of the mass of the original star, so a pretty solid chunk. But that number is pulled out of my ass—I am not a physicist.

But there are even more weird effects going on, due to the warping of gravity the mass of neutron stars can be up to 20% less than you'd expect based on its baryonic (neutron) constituents (questions 4 and 7):

https://www.astro.umd.edu/~miller/teaching/questions/neutron...