Rokus don’t consume much power and many recent ones have an (optional) power saving mode where they go to sleep after a period of inactivity, so they’ll consumer even less power. In general, I think most people would recommend returning to the home screen after watching, not just pressing pause.
There are several models labeled “Roku HD” and they all came out 10 or more years ago, so they may not have the power saving mode. (However, I don’t think that anything labeled “Roku HD” was in a stick format. This is why we tend to prefer the 4 digit model numbers – Roku likes to reuse the same or similar names over and over.)
Next time you see your Roku as “off”, try pressing the HOME button to wake it up. If that fails, try turning off the power saving feature.