Definition of temperature and its relationship to relativity

So I've been thinking about temperature and its relationship to the root mean square velocity. This is just some thoughts and an attempt at an open discussion.

In researching the issue, I have read that root mean square translational velocity is only part of temperature which really is determined by oscillation around all degrees of freedom, not just translational degrees of freedom.

I was thinking, it seems extremely counterintuitive (as this paper also agrees ) to discuss temperature as anything other than lorentz invariant.

At first glance, it seems like taking the root mean square velocity of the material as a component of the temperature, it would imply that you could have a glass of water frozen to one observer, but boiling in another frame of reference--unless you also presume that boiling points and freezing points depend on reference frame as well.

If temperature weren't lorentz invariant, it would imply that stars moving quickly towards or away from us would both appear hotter than they are in their own reference frame. I'm not sure which would be larger--that or the doppler effect. Either way, it seems like we'd consistently measure stars further away as hotter than those closer, as the expansion of the universe would put them at more relativistic speeds compared to our reference frame.

Now, if we talk about temperature as lorentz invariant, it seems like there's an easy way to reframe the whole root mean square velocity to make it more intuitive that it is lorentz invariant. Just call it the standard deviation of velocity instead of root mean square. The math would be the same, right?

But then you have the issue of generalizing that to the other oscillators. So with rotational oscillation, would taking the standard deviation of that oscillation also make more sense as a way to describe the temperature?

If you have a bunch of particles with a lot of the kinetic components in the form of rotation, but that rotation is all in the same direction at the same angular speed, does it make sense to say the substance has a lower temperature than if they are rotating in random directions as random angular speeds?