An upper bound in temperature?

[fluidistic]

I'm wondering if a gas in which all its molecules are moving very close to the speed of light has a finite temperature.
More precisely, if we take the limit of the speed of the particles to be exactly the speed of light (I know it's impossible to reach, but as I'm calculating an upper bound I think I can do that), is the temperature of the gas infinite, as the kinetic energy of the molecules and the masses of the molecules, or is it finite as their speed?
If it is finite then there exist an upper bound temperature for any body. In this case, does the upper bound depends on the nature of the gas? (for example an electrons gas, a protons one).
Thanks.

 

[JoAuSc]

Roughly speaking, the temperature depends on the average energy of the molecules, not the average speed, so there would be no upper limit.

 

[fluidistic]

Roughly speaking, the temperature depends on the average energy of the molecules, not the average speed, so there would be no upper limit.

Ok thanks. I got the confusion since I've heard in my class that we could define the temperature of a gas by the number of collisions of the gas' particles against a wall per unit of area divided by one second.