(There is another constant in the formula that depends on what definition of mean you use, but it's safe to ignore it for this discussion.)
So if T is big enough, the result of this formula is faster than light.
But this formula is useful only for a not relativistic gas. Once the temperature is so big that relativistic effects are important, you must use another formula. (The other formula is more difficult to calculate, but when the temperature is low the result is almost identical to the formula I wrote.)
v ~= Sqrt( k_B * T / m) ~= constant * Sqrt(T)
(There is another constant in the formula that depends on what definition of mean you use, but it's safe to ignore it for this discussion.)
So if T is big enough, the result of this formula is faster than light.
But this formula is useful only for a not relativistic gas. Once the temperature is so big that relativistic effects are important, you must use another formula. (The other formula is more difficult to calculate, but when the temperature is low the result is almost identical to the formula I wrote.)
Temperature has no theoretical upper limit, but if it's high enough weird things can happen as described in a sibling comment. More details in https://en.wikipedia.org/wiki/Planck_units#Planck_temperatur...