I cant see how cycling on and off would be any issue with an LED (how many thousands of times do leds in a lightbar turn on and off in a matter of minutes).
Cheaper prices come with cheaper build and QC. Even though it is below freezing, the diodes still need to desipate a lot of heat (even at -30f, my engine would overheat if the coolant stopped circulating). If they do not have a good heatsink or attached to it properly they could overheat. Could also be the solder/electrical connections inside failing (from heat, corrosion, poor manufacturing or vibration).