Since you have mentioned that it is a thermal paste problem,
Did you monitor your system temperature whether it is stable or not stable?
When running system scan for viruses you are actually pushing your CPU consumption to 100% thus increase the temperature of your processor.
Once your processor reach the auto shutdown temperature, of course your system will shut down to prevent overheating and damages to your system hardware.

If you does not encounter blue screen, then please send your system for thermal paste check.
If you does encounter blue screen, please do upload your minidump file here so that some of us can help you.

But before any of this, try monitor your system temperature using application like Speed Fan or Everest

Thanks

Send it to shop and let the technician open your processor and fan to check whether there is a need to put a new layer of thermal paste.
As for now, just monitor your system temperature and see the different when you run a system scan and when the system is idle.

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You are using a P4 processor.
If that is the idle temperature for your system,
it is consider very high.
Further when you running system scan, you are pushing your system.

Normal temperature for a P4 should be around 40 - 50C.
Even my PD normal temperature only 55C.

Please do apply a new layer of thermal paste to your processor.
And,
Please do clean up your system from any dust for better air flow. Especially those crucial cooling fan.

Thanks