Are you sure it is ashServ.exe as that is the scanning engine and not the avast update process, avast.setup.
I have never see ashServ.exe connect to the internet, nor scan internet content, which the ashWebSv.exe does.
So I’m somewhat surprised that you say it has connected to the internet, this is an example of my firewall log showing shortly after connecting to the internet (dial-up) what activity happens relating to an avast update check.
I have seen it on this computer when avast was first installed. Since it is set to block, it hasn’t tried, at least the firewall hasn’t notified me of any attempts.
This is just a PING packet, finding out if the Internet connection is alive. You can turn this off by checking the “My computer is permanently connected to the Internet” box in the avast settings, Updating (Connections) page.