The server it is trying to reach is the Alwil server from which it can get updates (if there are any). Very strange…

I think this is something for one of the people from Alwil, but till they jump in, we can gether some more info.

What is the setting under “update (connections)”?
Does the behaviour changes if you change that setting?
Does it still wants to connect after a reboot?
Does it tries to connect every 4 hours?
Does it still happen after a repair of Avast?
What exact version of Avast are you using? (right click the blue ball > about)