Is temperature quantized?

I'm learning quantum mechanics on my own. I've known that energy is quantized and I've started wondering about temperature. From thermodynamics we have: $$U=\frac{3}{2}NkT$$ (for ideal gas, of course)
Both U and N aren't continous, so i think T shouldn't be, too. Is that formula correct also for quantum mechanics?
I'm really sorry because of my language, I'm still working on it.

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 In the canonical case it's clear that $T$, if coming from outside, can be continous (the probability \$0