Why is Fuzzy Dark Matter Cold? Fuzzy dark matter (FDM) has a typical mass of $10^{-22}$ eV. With such a low mass, why is it typically assumed to be cold? That is, what keeps the FDM non-relativistic. With such a low mass, wouldn't almost any energy input cause a FDM particle to have a momentum significantly larger than its mass? Or is it assumed that it formed cold and then doesn't interact at all over a timescale longer than the age of the universe?
 A: Long answer short: Yes, it is assumed to have formed with very low energy ("cold"), and further, that it continues to be decoupled from the rest of the Universe so much that it stays that way. Any dark matter model that predicts such very light particles but with different formation mechanisms (e.g. the common thermal production) or non-negligible couplings is ruled out by data.
Going through the questions individually:

With such a low mass, why is it typically assumed to be cold?

Any viable dark matter model has to be "cold" to agree with data, specifically with structure formation. So it's assumed to be cold to be a viable model in the first place.

That is, what keeps the FDM non-relativistic. With such a low mass, wouldn't almost energy input cause a FDM particle to have a momentum significantly larger than its mass?

Indeed it's somewhat counter-intuitive that such very light dark matter would be low-energy enough, and would remain so. In the limit where your model decouples this dark matter completely from the rest of the universe that's not a problem though.

Or is it assumed that it formed cold

I'm not aware of a model where it isn't formed cold in the first place. Some sort of a "Mis-Alignment Mechanism" is a popular choice. If you were to form it hot or warm, /and/ have it decoupled from the rest of the universe such that once it's cold it stays cold, it's hard to see how it could cool in the early universe. Thus: make a model that forms it cold in the first place.

and then doesn't interact at all over a timescale longer than the age of the universe?

Right.
Somebody in the comments mentions boson condensates. That is a possibility but not necessary. For axions in particular there's a long debate whether they would form a condensate or not; I think the bottom line is simply that it depends on the quirks of your model whether you get a condensate or not.
