This behaviour is either caused by application you have installed (not likely it is Avast) or a broken motherboard.
Does the time in the bios stays accurate? If not, your motherboard is broken.
A battery loosing its power (running empty) does not slow down the clock. It either keep the settings correct or loose them.
If it is not software (mostly it isn't the software), than it is the circuitry on the mobo that handles the time (chrystal, and some other electronic parts) that is breaking down.
Easy way to test.
- Create a win98(se) bootfloppy.
- Boot to the bios.
- Set time/date correctly
- Reboot and use the bootfloppy
- Leave the system on for about 72 hours (this is needed to avoid that you won't notice a very small change in the time, depending on how bad it is, you can see a time shift earlier)
- Check the time
- If it is not correct anymore, your motherboard is broken