There are a few ways you could do this.
You could set up a sync function in Sync Gateway that assigns the document to a special channel whenever the value for that key changes. You'd then do something like run a filtered replication that listens to that channel, and use a change listener to notify your app.
You can also do this with a live query. The most explicit way would be to create a View with a map function that emits that key/value as its index. Then just set up a live query to get notified of changes on the view.
You could also set up a general database change listener and check for the value to change directly yourself.
I can think of other ways to do it, but those seem like the most sensible. The efficiency, and which you pick is going to depend on specifics of your application.