LocalBroadcastManager has been deprecated. What I should use instead in its place?
LocalBroadcastManager is basically an event bus with a lot of unnecessary ceremony around Intents and intent filters. So one replacement is easy, and functions quite similarly: you can use any event bus library. greenrobot's EventBus is a popular choice (here's a guide for it) and Guava also has one, if you're already using Guava (but Guava is pretty heavy to include just for an event bus).
But event buses suffer from the same problems that LocalBroadcastManager does that led to it being deprecated: it's global, it's not lifecycle-aware, and as your app gets larger, it becomes much more difficult to reason about the effects of a change to an event. For cases of observing data,
LiveData solves this quite nicely because it's lifecycle-aware, so you won't get change notifications at the wrong time (like before your View is set up, or after
onSaveInstanceState) - but it'll handle delivering the change notifications when you're in the right state again. It's also more tightly scoped - each piece of LiveData is accessed separately rather than having (typically) one event bus/LocalBroadcastManager for the entire app.
For cases where it's more of an event rather than a piece of data being changed, you can sometimes convert it to a piece of data. Consider if you have "login" and "logout" events - you could instead create a LiveData that stores an Account for logged-in users, and becomes null when the user is logged out. Components could then observe that.
There are certainly cases where it really is difficult to convert it to a piece of observable data (though I can't immediately think of any examples that would typically be used with an event bus patten). For those, consider writing your own listener interface, similar to how on-click listeners work.
For your example of a countdown timer, I think LiveData is a pretty straightforward solution, and will be much easier than an event bus or even LocalBroadcastManager would be. You can just have a LiveData of the timer's current value, and subscribe to it from whatever needs to show the value.