I have been having what appears to be overheating problems in an HP a1230n Pavilion with a stock Athlon processor.
If I keep the case open and turn a fan on the inside of the case the computer runs. If I do not do that, it locks up. Ambient (room) temperature is 84 to 85 degrees F.
The application SpeedFan indicates the processor is running with a core temperature that is acceptable. The "remote" temperature closely parallels the core about 30 degrees above. So, the core will read 84 degrees F and the remote will read 114 degrees F.
If the remote goes to 128 to 134 degrees F the computer rapidly (within 5 to 10 seconds) locks up.
I installed a new heat sink and fan. The interface between the CPU and heatsink is Arctic Silver 5. I put it on very carefully after cleaning the surfaces and prepping them according to Arctic Silver's web site.
If it is possible that I have a bad CPU, I have been thinking of upgrading to the best processor available for this computer. That might bring some stability and gain speed as well.
I have been fighting this for about a year. No money for a new computer (and too stubborn to give up till this is solved).