Why can I hear noise (white noise / pink noise / brown noise), but not hear temperatures?

EDIT FOR CLARIFICATION Air temperature is caused by air molecules moving randomly at high speed, white noise is caused by air molecules moving randomly at high speed, what's the difference? Why does white noise fill the room with sound instead of just raising the temperature slightly?

My hand-wavy-sounds-like-science-technobable guess is that temperature does fill the air with sound, but most of the energy of that sound is at far too high frequencies for my eardrums to detect (in part because my eardrums are emitting noise at a those frequencies). Maybe the average wavelength of thermal noise is roughly the mean free path length of the air molecules, so the average frequency of the noise is roughly 5 GHz. But I just making stuff up and really don't know.