If I have an LED which is specified as, say, 350 mA at 12 VDC. What would happen if I fed it directly from a 48 VDC constant-current power supply providing 350 mA?

I understand increasing the voltage on an LED typically makes it run hotter as it tries to draw more current, but if it is fed by a current limited source is this issue voided?

Edit: I am talking about higher-powered LED's, like ones used for outdoor lighting such as streetlights, garden lights, etc. Say a max. wattage of 30 W.