Whee, the best thing to say about "... just because something is moving relative to another thing does not necessarily mean it is hot ..." is that this is unsettled. ;)

Ott 1963 shows T' = T / {\sqrt{1 - v^2 / c^2}}, or for short T' = \gamma T, where \gamma is the Lorentz factor, T' is the temperature calculated by an observer in a set of coordinates Lorentz boosted with respect to the coordinates in which the system is at rest with a temperature T.

Asymptotically one would expect that as \gamma -> 1 \RightArrow T = T' and \gamma -> \infty \RightArrow T = \infty assuming one is measuring temperature using photocalorimetry (consider a blackbody radiator...).

However, there are different proposals that depend on how and if one makes the first or second laws of thermodynamics Lorentz covariant, and how one goes about measuring a distant object's temperature. All of this is also in the context of Special Relativity; a general relativist could say that the whole matter comes down to a choice of gauge anyway, and temperatures are only directly comparable for two objects at the same point in spacetime (and even there you get a can of worms because they would then be "in contact" at the point p on the manifold, and even then there is a choice of gauge and some interesting cut-offs in which they could equilibriate (T_obj1 = T_obj2), dissipate excess stress-energy (T_{p} > T_obj1 + T_obj2), or collapse gravitationally).

Unfortunately the matter is not wholly settled and there is an absence of firm experimental evidence, and moreover the debate is confined to flat spacetime.

However, one has to do quite a bit of twisting to assume that T = T' (Landsberg did this in 1966 & 1967).

Here is a nice recent set of slides on temperature in GR: http://www.dpg-physik.de/dpg/pbh/aktuelles/pdf/Andersson.pdf

Q. Is a moving body hot/cold? A. Yes/no/maybe. The discussion is "pointless".