If you switch your system off between usage, that may be what is triggering it to check again.
Whilst your system may be old there are many with much lessor systems without issue, so generally it should be up to the task. You could check what other programs are allowed to run on boot and be severe on what you allow (security based software, system services, absolutely essential programs only).
How long have you had avast and or was this previously OK ?
Have (or did) you another Anti-Virus installed in this system, if so what was it and how did you get rid of it ?
The Update check interval is 240 minutes, this can be increased - avastUI > Settings > Updates - 'Auto-update interval' section and change the value, increase to say 480. That however may, not help on the first check after boot, if not there is another option which we can try if required.