I have a Roku in the living room, and Roku+ in the bedroom. I was using the standard ATT Smart Home Manager to see what's what, the bedroom Roku+ is listed as having the most "activity" every hour of the day. We don't even use it. I used nmap to watch what it's doing, and it's mostly the router sending a "who's using IP addr so and so", and not getting an answer.
So I think something about the Roku+ is causing constant router messaging to it. I guess that doesn't have anything to do with the internet though, if it's just between the router and the Roku, but I don't know.