This reminds me of a bad beat I got dealt with pocket AA that I lost because I didn't bet hard enough on the flop.
There were two all-ins before me. I called one all-in and raised another, with one other limper who called me.
The flop was a rainbow. The limper made a $2 bet and I called. (BIG MISTAKE thinking I could smooth call on over pair).
On the turn, he makes another $2 bet. I call him. (another stupid mistake).
The river comes. He bets and I call. He makes trips on the river (heraised holding JTs). I should have just went over the top with him on the flop. He only flopped a pair of tens.

Hands like AA, I tend to bet big. It also depends on what kind of raises you can make preflop that will get people to fold. I minimum raise preflop with AA isn't enough. Maybe 10-15x BB?

If you flop trips with AA (or any pair), and the flop looks scary (two cards to a suit/two cards to a straight), bet big... Sometimes you HAVE to push all-in, especially if you have the nuts on the flop. More times than not, someone will call incorrectly and lose.
Those eejits who call your AI or your raise with a flush/straight draw make me cringe, especially if they get a runner-runner.
I went on tilt for a couple of days because that happened.