I recently heard a story from a friend about a pilot that broke a TFR which was issued after takeoff. This sounds suspicious to me that the pilot had no advance notice of the TFR. Is the FAA able to issue an immediate TFR with no advance notice? If so, how can a pilot become aware of these?
The FAA can issue TFR's on short notice, and if you are flying a slow plane or a fast plane far they may very well pop up while you are inflight. While things that cause TFR are often planned in advance like the super bowl there are plenty of last minute things like a wildfire or a quick movement of the President that will force a fast TFR to "pop up". Even the FAA admits they can can be unexpected
TFRs can pop up unexpectedly, so before any flight, be sure to research NOTAMs, check the TFR list online at http://tfr.faa.gov, and call Flight Services for updates at 1-800-992-7433.
I can not find any regulations sating the FAA has a minimum amount of notice time they must give.
A pilot flying VFR in an airplane not radio equipped is always at risk of busting one of these. But if you are in a radio equipped aircraft taking flight following or monitoring your local FSS is a good way to be aware of these kinds of events.
This AC covering TFR's does not list a minimum amount of time they must be filed before becoming active.
There are multiple types of TFR (see 14 CFR 91.137 to 91.145) and in at least one case, the regulations allow a TFR to be established "immediately" (emphasis mine):
§91.139 Emergency air traffic rules.
(b) Whenever the Administrator determines that an emergency condition exists, or will exist, relating to the FAA's ability to operate the air traffic control system and during which normal flight operations under this chapter cannot be conducted consistent with the required levels of safety and efficiency—
(1) The Administrator issues an immediately effective air traffic rule or regulation in response to that emergency condition; and [...]
Two simple (?) ways to be become aware of pop-up TFRs in flight are to be talking to ATC, and to have ADS-B In.
If you're on an IFR flight plan or flight following then ATC should inform you about TFRs before you fly into them, although that isn't guaranteed. And ADS-B In data includes TFRs, but I have no idea what the refresh delay is from a TFR being issued to it reaching the cockpit.