Entanglement is about states. Objects can carry such states but their other properties / states are in fine independent from the entangled states.
Let's now consider heating and for example two entangled polarization states.
Does heating the object carrying the entangled state on one side, lets us get a kind of polarization measurement ? I don't find how...
If you assume no , then you can conclude that they are unrelated and heating one from the pair doesn't affect the other.
Else , if you assume yes, what would happen ? the other spin state will adjust its polarization in a statistical way consistent with QM but I don't see how we can say something on the heat on this side.
Black body radiation of both objects behaves as usual.