For that you need a service, not an AsyncTask:
http://www.vogella.com/tutorials/AndroidServices/article.html
Services persist even after activities are dismissed. Be careful though, draining the battery will anger your users. Also, services can be destroyed if memory runs low, so be careful.
The scheduling of a task to run in the future should use AlarmManager. The execution of the operation should run in a service since you specify that sometimes it will occur when the app is not active.
You can also use a TimerTask but it's a little more drain on battery and not as precise as AlarmManager. You can also use a handler to postAtTime
Here is a good reference:
difference between timer and alarmmanager
and another one:
What is better in Android? Timer or Alarm?
It really depends on the frequency and use-case. I've used all of these - I view TimerTasks as "keep checking frequently for short time" vs. handlers for "check again in a while" vs. AlarmManager for "I want to keep checking every hour for a long while"