How to implement time expiry hot observable in RxJS (or general in Reactive Extensions)

All you are missing is to schedule a task to replace your cachedData with a new AsyncSubject after a time period. Here's how to do it as a new Rx.Observable method:

Rx.Observable.prototype.cacheWithExpiration = function(expirationMs, scheduler) {
    var source = this,
        cachedData = undefined;

    // Use timeout scheduler if scheduler not supplied
    scheduler = scheduler || Rx.Scheduler.timeout;

    return Rx.Observable.create(function (observer) {

        if (!cachedData) {
            // The data is not cached.
            // create a subject to hold the result
            cachedData = new Rx.AsyncSubject();

            // subscribe to the query
            source.subscribe(cachedData);

            // when the query completes, start a timer which will expire the cache
            cachedData.subscribe(function () {
                scheduler.scheduleWithRelative(expirationMs, function () {
                    // clear the cache
                    cachedData = undefined;
                });
            });
        }

        // subscribe the observer to the cached data
        return cachedData.subscribe(observer);
    });
};

Usage:

// a *cold* observable the issues a slow query each time it is subscribed
var data = Rx.Observable.return(42).delay(5000);

// the cached query
var cachedData = data.cacheWithExpiration(15000);

// first observer must wait
cachedData.subscribe();

// wait 3 seconds

// second observer gets result instantly
cachedData.subscribe();

// wait 15 seconds

// observer must wait again
cachedData.subscribe();