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[Black Aeronaut]

The colder the air, the more dense it is, the more you'll suck into the engine. The more air you have in the engine, the more power you get. I think.

Also, it's not just cooling the intake air, it's heating the liquid hydrogen fuel to the point where it'll turn to gas quickly when injected into the engine bell.

Ahhh, so that's the coolant.  Figures.  You know they used the fuel itself on Blackbirds as the hydraulic fluid?  Yeah, their way of cutting down on weight - the stuff's got an obscenely high flash-point anyhow.
As for denser air... I could see that, but I would think that there would be workarounds to compensate for it.