for now, I check the activities when my app become foreground, but I can't be notified when my app is in foreground and server start an live activity by push notification. I've tried "content-available": 1 in LA's payload aps, but I also can't be notified in the didReceiveRemoteNotification in appdelegate.

Yes, you mean my app can be woken when Activity.pushToStartTokenUpdatesBut delivered the new token. But I want to know how can I get the activity.pushTokenUpdates when my server start an live activity via push notification, because without the activity.pushTokenUpdates, my server can't deliver update information after start the live activity.