Do you use any battery optimizers? Those are known to prevent Aquamail from syncing messages. See. e.g. FAQ (the last item):
http://www.aqua-mail.com/?page_id=227 .
As an aside: It is not clear what type of connection to the server you have: POP3 or EWS (since you don't have IMAP), and if it is at one of the major e-mail providers (Gmail, Yahoo, Hotmail, etc.)
While this will not affect your syncing, you would benefit from using IMAP type of connection as opposed to POP3, if it is POP3 and if your provider supports IMAP (most do).
You can read in the same FAQ about the differences.
And while I agree with Paris Geek's suggestions, - I'd suggest to use the frequency of syncing at either 15 or 30 min.