Edit: Ok it turns out there's 2 parts to a cooling roof, how reflective it is, and how good it is at emitting infrared radiation (emissivity). Reflectivity of visible and UV light prevents more energy from being absorbed, but doesn't necessarily mean the material cools down well once hot. A high emissivity means the material can cool down quickly through radiation.
I guess YInMn blue has a high emissivity.
https://en.wikipedia.org/wiki/Reflective_surfaces_(climate_e...
Rare earths are not going to come down in price, in my opinion. They're used in too many places and have too few sources. The US has a rare earth strategy document that makes clear just how insecure supplies of these elements is.
Is it also white to the x-ray spectrum and the radio wave spectrum? If so, I am pretty sure that this doesn't actually exist.
Easy to ignore since it won't be hit by any.
> radio
I dunno, how hard is it to get materials that reflect radio waves?
It's probably reasonable to talk about materials that reflect almost all the light in frequency bands of non-negligible power as effectively being actual white.
More useful ones being things like neodymium and erbium and cerium.
Once one of them is not being produced in sufficient quantity due to being a byproduct, then it gets really expensive. That's for sure.
(edited for clarity)
Personally I'd probably focus on windows and solar insolation from that first, especially since the roof is insulated, but that makes more sense.
It'd likely be more effective to do some combination of insulation, radiant barriers, and thermal breaks.
So yes, synthesis, quality control, probable patent encumbrance, and high demand, more than raw ingredient price: but it'll never be particularly cheap.