The LED's are not powered by the Arduino. It is wired on a breadboard, like so:
PSU (12v, 1A) -> voltage regulator (STM 78S05 2A) -> 5v
5v -> LED's
5v -> Arduino VIN
Arduino D3 + D5 -> LED data input
All share the same GND. The voltage regulator is getting hot, the PSU is not.