I just bought the Avast premium package and i also notice, that my HDD (just a data drive, my system is on a SSD) is running for 2-3 minutes.
I didn't check the system load, its just annoying, since its "loud".
This only happend after activating all the features from premium and doing all the recommended scans and things.
This did never happen with Avast free, but always since premium, so some "service" is doing something at each start, like a scan of the HDD.
I couldn't find anything in the options about a scan at each start of windows though.