The badge is usually cached, but since it's updated frequently, it is not written to the on disk cache except when Arctic is closed. The badge is supposed to update automatically when launching Arctic, or changing accounts. However it looks like at some point I removed the initial update on launch. So the badge still updates, but not for about 30 seconds after opening the app, or like you noticed when opening the inbox tab.
I'll fix this in today's update to add the initial update back in so it will fetch the unread count immediately.