Thanks for the explaination, but which operation could that be on the Gmail-Account at 9:10? The Gmail and the Hotmail are Push only and there was no new mail at 9:10 on the Gmail. Is there something like "checking mails" even on Push? I always thought Push is only server sided and the client is only listening/waiting without any active checking.
Under settings / network there is an option for "Push mail - keep alive". Maybe the shown time under the account name has something to do with that? (new packet for keep alive sended?)
A few thoughts on that:
1. My understanding is that depending on the server, "Push" may require regular re-establishment of that connection. (I believe "keep-alive" is helping to assure the connection is not lost for that reason.)
2. Moreover, I don't know if Gmail servers have some time limit for the "IDLE" connection being open. They might, - to enforce effective load balancing.
3. Also, IDLE assumes a sustained open TCP connection. as far as I know, it will not survive roaming, i.e. when your device moves to a different network and obtains a new IP address, that connection has to be re-opened.
4. Occasionally the IMAP server can be rebooted momentarily (e.g. to replace the SSL certificates on the server, which Gmail does weekly or biweekly now), in which case the connection has to be re-established.
All those events (I am not 100% sure about #1) will be activities that are likely to trigger Aquamail to show the last activity time.
BTW, as recommended by the developer, even with PUSH, you do want to keep scheduled sync enabled (say at every 30 minutes). That will ensure you will not miss any messages.
HTH.