I’ll start by saying I’m not a physicist so I can’t say anything about relativistic effects, if that’s what you’re interesting.

In fluid mechanics we talk about “stagnation properties”. This can refer to a variety of fluid properties, commonly pressure and density, but most commonly temperature. Stagnation properties are defined as the value of a certain property when the fluid is brought to rest. For pressure and density this requires the fluid to be brought to rest isentropically (no entropy rise) and adiabatically (no heat addition or removal), but temperature only requires it to be adiabatic.

Using stagnation properties results in concepts of static and dynamic properties - the stagnation property is static + dynamic. A static property is the property that a fluid particle experiences when it is travelling at the same velocity as the fluid i.e. it is static relative to the fluids. For instance water in air will start to condense when the local static temperature and pressure fall to the dew point. A dynamic property is the component of the stagnation property due to the relative kinetic energy. This means that, yes, the temperature perceived by an object depends on the speed it is travelling at relative to the fluid. At high altitudes the temperature may be 230K, but an aircraft flying at high speeds experiences a stagnation temperature of maybe 260K (the exact value depends on the Mach number).

Temperature is a statistical quantity that happens to be proportional to average energy in a system of particles following a Maxwell-Boltzmann distribution. However, it is not proportional to average energy by definition. If the gas has a non-zero average velocity in your frame it is not following an MB distribution, so it's temperature is not proportional to its average energy in your frame.

As a side note, you would observe blackbody radiation that was red- or blue-shifted depending on your motion that could make the gas appear warmer or cooler.

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No; the kinetic energy corresponding to the temperature is measured relative to the center of mass. A cold body moving fast doesn't appear hot, as the relative undirected motion of the particles is unchanged. (However, two cold bodies colliding inelastically would get hotter, of course.)

> As a side note, you would observe blackbody radiation that was red- or blue-shifted depending on your motion that could make the gas appear warmer or cooler.

A hotter or colder body's blackbody radiation isn't simply shifted by a set amount, so this isn't true. You'd identify the same temperature with some overlaid bulk motion. I apologize; my statements were incorrect.

One annoying thing about blackbody radiation is that is still looks like blackbody radiation after Doppler shifting. It's why the CMB has a blackbody temperature of 2.7K, even though it is coming from ionized hydrogen.

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Your answer seems to imply that if the system was spinning, you would call it higher temperature, because you can't "remove" the motion by going to the center of mass frame. I agree that it's useful to gasses to go to the center of mass frame to restore the distribution of velocities to a Maxwell-Boltzmann distribution so it looks more like a system with a well-defined temperature. However, I don't think there's anything in any reasonable definition of temperature that says "it's measured relative to the center of mass"

> Your answer seems to imply that if the system was spinning, you would call it higher temperature

That would be a misreading, because the context of the answer is a question about translational motion. More generally, the bulk motion is typically subtracted before we do thermodynamics. If you don’t see that stated in definitions of temperature, it’s because it’s already been implicitly assumed.