There are plenty of temperate and cold water corals, but you won't find them for sale in the trade.
The only folks I have known who have run cold water setups have done so specifically for the divine Catalina Goby -- which, alas, you DO find in the trade sold as a tropical fish instead of the temperate fish that it is.
Lighting would or would not be an issue based on the specific species kept. The temperate waters off the coasts of CA for example get tons of sun, but the water is still pretty cold. Deep and very cold water corals like off the coast of Alaska need very little light.
A serious chiller, however, would not be an option!