> If it could heat up, Dark Matter wouldn't be dark.
I don't think this is correct at all. A system doesn't have to interact with EM to be thermodynamic. If you can define a temperature for it and there is the possibility for energy transfer, then it can heat up
I think this is where the debate is. I’m not a physicist but my understanding of the current dark matter models is that it doesn’t interact with itself in a way that could be thought of as “energy transfer” (ie. like particles that collide), but only gravitationally. This would mean there’s no real way for a dark matter “particle” to transfer momentum to another particle, and thus no real way for “heat” to exist as such.
> If there is any way to transfer energy, then it can heat in exactly the same way as anything else.
If it could heat up, Dark Matter wouldn't be dark. So yeah, dark matter that heats isn't quite what one would expect.