If you have been messing around with your cooling system before your temps started shooting up on you, you may want to double-check to be sure you have the cooling fan plugged into its header properly. If your plug is a 3 pin plug and you're plugging into a 4 pin header you could have missed the proper alignment, and are plugged into the wrong 3 pins. of course, if any of what I just said is true simply looking at your fan to be sure it's spinning would be a good indicator. If you're using an AIO setup, it can go unnoticed, so the temps would be a good indicator as well as the RMP gauge for your water pump. In my case, I have an issue where I'm on the correct 3 pins of a 4 pin header, but just a slight wiggle or bump of the wires leading to my water pump plug will cause my water pump to stop pumping water. I think one of the pins could be bent somewhat, and causing a loss of connection on that pin in the Molex. I have to be careful whenever I open my case to check my gauges afterward, if I don't, my CPU temp will shoot up so high it will shut the CPU down without any warning what so ever. I'm using an ASUS motherboard at the moment, and it's motherboard software does not have an alarm that I can set for high temp. Well it does have a setting, but it's doesn't sound an alarm if I set the limits on it, it only shows that reading in red, so I have to have the software running and looking at the numbers in order to know. The UEFI BIOS has an audible alarm,, but only if I am booting up will it warn me, if I'm already in WIndows it'll not trigger. My MSI motherboard I had prior to this ASUS board, had alarms I can set for any measured item that it reads from the motherboard, and it'll go off in Windows. I'm surprised the ASUS software does not. They need to catch up with MSI in this regard.
Also, be sure you have applied enough thermal paste between your cooler and your CPU, since you mentioned you just replaced your cooling system, it's possible you may not have re-applied any new paste between the two.