I think that the term
is invariant. Because
is the velocity of the molecule
relative to the box which is invariance, no matter what frame of reference you are.
I think we have a different reasoning here.
The answer for the "paradox" that I have proposed above is that my explanation really contains "hidden" flaws.
First of all, when the observer is at rest with respect to the box, he sees the molecule
has velocity
(This is why I was at first confused when I read Mwit_Phychoror's explanation because
, in my definition, is not viewed by the moving observer but is relative to the
box)
From this point, one can prove, by applying Newton's law, that
Now let's consider the observer who moves with velocity
relative to the box. (For simplicity, let the direction of motion of the observer be perpendicular to one side of the cube-shaped box.) He sees the molecule has new velocity
and new average velocity
. There has been no fallacy until this point.
Here comes the flaw. We
cannot apply
by plugging new average velocity in this equation because this equation is proved by the fact that the box is not moving with respect to the person who utilize Newton's law.
To be clear, let's prove kinetic theory when the cube (box with dimension
) is moving with respect to the observer to the left with velocity
. When we measure the change in linear momentum in x-axis direction of the molecule approaching the left side of the cube, we write
which illustrates that the change in momentum remains the same regardless of the motion of the observer.
Now the time elapse
of the first and second collisions of the molecule with the same side of the cube
is the same as that measured by the other observer who is at rest relative to the box because time interval between two events, in classical model, is invariant.
Thus we ends up with
not 
, which indicates that the anticipated temperature is the same regardless of the observer or, in other words, frame of reference.
Is "absolute" temperature still an absolute quantity? At this point of view, I'm not hesitant to say yes. However, we prove this from the classical approach. There could be more complexity in modern model, unfortunately, that I still don't know for now >o<"
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