It could be a theme or widget which runs as an app, like my ICS Status bar app which replaces the stock status bar in GB. Since its an app, if the phone is left idle and Locked for a time, the system closes it and I have to relaunch it. In my case at least, I'd like that one in particular to keep running.
In a pinch, if you're rooted, download and install root explorer and uninstall your app and then download its apk file online and then use root explorer to move the apk manually to the /system/app folder (DON'T INSTALL IT) then reboot. The app will self-install as a service vs. a normal app and therefore usually remain active
Samsung Galaxy Precedent (rooted, AreaRom 3.4 , sd-ext w/ 3.65GB internal space)
Kindle Fire (1st Gen, rooted, AOKP JB 4.1, Nova Launcher)
Samsung Galaxy SIII 4G LTE (rooted, Carbon ROM)