Surprisingly you can plug in a thunderbolt cable into the displays back ports instead of using the cable connection. I would give that a try to see if the issue is the cable.
***Overheating is often a power issue! Either the houses wiring or forcing a bad connection setup.***
Make sure you are using the same outlet as your system for the display using a surge suppresser strip or UPS.
One of the risks is bad grounding of the outlets and/or inverted hot (Black) with the neutral (White) wires of the outlet or plugging the device cable into the outlet upside down (broken ground pin or using a three to two blade adapter or using a two blade power cord).