This is a question I've tried to search for an answer since quite some time but never found any so far.
It is happening on a stock HSPA+ Galaxy Nexus with JRO03C 4.1.1 Jelly Bean (bought from US play store).
Please note this behaviour was happening with Ice Cream Sandwich as well.
This happens with more than one app, but this time I'll focus on play.google.com/store/apps/details?id=com.merriamwebster
because it's free and I was able to reproduce this behaviour quickly.
Steps to reproduce:
-Open up Merriam Webster's app
-Write any word in the search box, than tap it in the drop down list that appears (you're brought to the word's dictionary page)
-Wait a few seconds then hit Android's home button on the bottom bar
Now check in Settings -> Apps -> Running:
You will see that the app is not on the running list but is in the cached processes list.
Now go to Developer Tools and enable "Show CPU usage" and hit home to show up the desktop.
You will see that com.merriam.webster is constantly using a small but not negligible percentage of cpu (its green bar is big enough to "highlight" the 'er' in .webster )
The only way to stop this is to stop the cached process in Settings -> Apps -> Running -> Show cached processes
I think the question now comes up by itself: "Aren't cached processes supposed not to use any CPU at all?"
Any info on the matter is appreciated.