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.
What is your forewall and or what reported it ?