For audaxes etc, I usually just leave it on the bike, switched on. That means that the total time keeps counting, and I can check the overall average speed remains above the minimum.
There isn't really any sort of pause mode on the eTrexes. You can just switch the GPS receiver part off - go to the satellite screen, press menu, then "Use with GPS off". This will stop the trip computer, and save some power, but you have to remember to switch the GPS back on again. So its probably quicker / easier to just switch the eTrex off fully.
Unless you want to look at the trip computer / map page etc while sitting in the cafe.