Quote from the Azure Web Jobs Documentation:
Persisted files
This is what you can view as your web site's files. They follow a structure described here. They are rooted in d:\home, which can also be found using the %HOME% environment variable.
These files are persistent, meaning that you can rely on them staying there until you do something to change them. Also, they are shared between all instances of your site (when you scale it up to multiple instances). Internally, the way this works is that they are stored in Azure Storage instead of living on the local file system.
Does that imply that by dropping app_offline.htm into the site root folder should pretty much bring down all instances simultaneously?