I was looking for cyan COB LED chips for replace the white LEDs from a solar fixture, for make it look mercury, but those are extremely expensive! A single 100w cyan chip can cost up to $50 while a white one cost less than $10. Why are those so expensive? Do they have rare and expensive phosphores? 

I do not know. I wanted to get a 30w cyan LED & swap out the warm-white in th LED streetlight I have, but it cost more than the whole fixture so I decided not to.

Niche product.
The LEDs are inherently complex to make technology, so all manufacturing process of a batch is expensive.
But it allows to process huge amount of units within the batch at once, so the cost get dilluted so the unit price could be rather low.

But with a niche product, you have low volume. The batch processing cost is nearly the same, but you sell only very few units, so each has to bear significant part of the high batch processing cost.

That makes sense, people barely buy cyan LEDs, so they are more expensive to make. The interesting thing is that amber LEDs are way cheaper than cyan ones, even if they are barely used.

Amber are used nearly in every signal light assembly: Traffic control, hazard warning, car turn signals, obstacle warning,...
Similar for red.
Plus wavelengths from red to green are way older and simpler GaAs based technologies than blue GaN based, so the processing is way cheaper.
The big difference is, you may made an actual wafer directly from the GaAsP based materials, but not from the GaN.
The reason is the crystallization of the GaN tend to develop so much of internal stress it will simply shatter on its own.
So you have to start from a monocrystal wafer of some other substance (Si, sapphire,...) with similar crystal properties as an substrate, only then grow just very thin epitaxial layers of the GaN, so it remains thin enough so the adhesion forces to the substrate are higher than the stresses within the GaN. Of course, the key is the selection of the substrate, its production and surface treatment to become usable as the substrate for the GaN growth. By far not simple process, comparing to other monocrystal semiconductor materials.
This indirect method took time to actually get developed into usable LED chip technology, there are even few Nobel prices related to exactly this...

The only reason the blue (so white) LEDs are of similar cost to the red/yellow/green is their huge market for general lighting, so even that very complex and expensive processing get that economy of scale cost advantage...

Now cyan needs the GaN (so the complex) process, yet has market way smaller than even the "tradditional" GaAs based LEDs (IR till green)

cyan needs the GaN (so the complex)

If you're talking about 'monochromatic', then maybe. But virtually all of the cyan (more commonly called 'ice blue' on the market, so try that search term, OP) Are just white LED with most of the phosphor missing. They're a bit more cost than white, but only because they're not used as regularly.

If you're talking about 'monochromatic', then maybe. But virtually all of the cyan (more commonly called 'ice blue' on the market, so try that search term, OP) Are just white LED with most of the phosphor missing. They're a bit more cost than white, but only because they're not used as regularly.

That could be, generally not much applications really need monochromatic light.
And so then it is in fact a blue chip (so the one already made in huge quantities for the whites, although of the GaN based chemistry) with a bit of green phosphor...
Still the low final production volume applies...

Not even green phosphor. standard white phosphor, just very little of it.