What may be the cause of hot start problems in a Lycoming fuel injected engine?
The issue is undoubtedly vapor lock. In the fuel injected Lyc 540, the fuel injection plumbing is on the top of the engine which is the cold side of the engine when you are flying but when parked, heat rises! The following procedure runs cooler fuel through the plumbing while flooding the engine, then the flooded start procedure is necessary.

First, check your POH for the hot start procedure for your plane.  If not there, try this as it is for a very similar airplane:

Hot Starts on the Saratoga PA-32-300:


Intentionally flood the engine.  This runs cool fuel through the plumbing and "breaks" the vapor lock.  


a) Full rich
b) Wide open throttle
c) Fuel pump on, watch fuel flow gauge
d) When gauge registers flow, wait 3-5 seconds then red knob to ICO and pump off.

Crank engine with red knob still at ICO and with throttle wide open at first. Gradually close throttle (at a rate that will take you about 10 seconds to close the throttle) and it should start up at some point.
When engine catches, do several things more or less simultaneously:


a) release ignition key 
b) throttle to idle to avoid over-revving
c) mixture full rich.