In the system of reference of spiraling on BH body its velocity is always increasing, so relative mutual motion of particles (=temperature) in the falling matter is likely to increase too.

After they disappear into a singularity, then if a singularity does not have internal structure, motion of what would we call a temperature?

Also there is another phenomenon related to BH: evaporation radiation has a black-body spectrum with certain temperature T. For Sun-mass BH it is in microkelvin range, for galactic-mass BH - in femtokelvin range.