For example, I don't monitor chat unless there are chat flags. I just use RSS feeds to monitor the main site and meta for new questions and pop in to handle flags a few times a day (usually before work, lunch time, after work, and times when I see a new post in my feeds and I'm free).
That's your call. If it's burdensome for you to run, I won't ask you to keep it running. But if it's not a problem, having an unfrozen room could be interesting. Although people may also be confused by it.