For Always On I can't think of anything where the compatibility level can matter.
However, for query execution there can be surprises. There have been many changes in the optimizer since SQL 2008, so many of your queries may get a new execution plan. In many cases, performance will improve, but you may not notice. However, there can also be regressions where one or two queries start to run painfully slow.
In order to prepare yourself for this, I recommend that you turn on Query Store, if you have not done this already. Let the database run for one or two weeks with Query Store on and the old compat level. Then flip the switch. If there is a query that regresses, you can force the old plan to solve the urgent problem.
Often when a query regresses like this, there is some problem with it. Maybe it is poorly written in some way. Maybe the indexes available are not the best for the query. With a little luck, the query ran well in the old compatibility mode, but such luck does not always last when you upgrade.
In SSMS, right-click the database, select Task, and at the bottom of this menu you find Database Upgrade, which helps you with using Query Store as I described it above.