Extreme temperatures, relativity and kinetic theory

According to kinetic theory, average kinetic energy is proportional to temperature. Supposing $k_BT/2$ per particle, can we use relativity and kinetic theory to calculate, e.g., the temperature and velocity of quarks in a quark-gluon plasma and hotter/denser states of matter?

Note: I did some calculations already myself to check this idea

Normal kinetic theory (in 3D)-> $$E_c(av)=\dfrac{1}{2}mv^2=3k_BT/2$$ must be substituted (???) by

$$E_c(rel)=Mc^2-mc^2$$

with $$M=m\gamma$$

For a quark-gluon plasma, taking the proton mass as m and the critical temperature as 200MeV I get

$$200MeV=3/2k_BT$$

so T is about $$2\cdot 10^{12}K$$ or about 4x10¹²K if I drop the factor 3 above. That is OK with the known temperature of the quark-gluon plasma. I am also concerned about the issue of determining the gamma factor for protons(quarks?) at those energies/temperatures...

I get, for protons at this

$$\beta=v/c=\sqrt{1-(mc^2/E)^2}=0.977$$

Am I right?