I thought a supernova's core temperature was absolute zero just before it collapsed, as it has run out of all nuclear and thermal energy, but I haven't been able to confirm this online anywhere and I was wondering if anyone can confirm this for me.
No, absolutely not. The core of a core-collapse supernova is one of the hottest places in the present-day universe. The temperature as the star runs out of nuclear fuel in its core is around 6-10 billion Kelvin. As it collapses, the core gets even hotter, perhaps as high as 100 billion Kelvin for a few seconds, before neutrino cooling starts to become effective.
We know that the temperatures are getting as high as this because at least two neutrino instruments on Earth detected roughly "thermal neutrinos" with energies of $\sim 10$ MeV coinciding with SN1987A in the Large Magellanic Cloud (see for instance Mirizzi et al. 2015).