I've got a 562XP which unfortunately doesn't have a warranty. It starts cold beautifully, it restarts hot just fine (slide up to choke on, slide back down to fast idle engaged). After cutting for a while it gets more and more likely that it won't restart, particularly if left for 10 minutes or so, and almost definitely won't start if I leave it for 20 minutes. Once I see that I have a problem, if I leave it alone for a while (never timed it) it comes good again, so it seems to get into some kind of temperature related state. Also, I've not sat down with a stop watch (yet), so my times may be a bit off.
Things that may be relevant..
When it's in this state the air purge bulb doesn't "work", pushing in feels too easy and it never pulls fuel through. I can tell when it's good to go again because I'll come back and the bulb will suddenly be almost completely full and it'll start just fine (though it does idle a bit funny after these restarts, sometimes dies but starts right back up again, but after the first cut it's back to normal).
Oh, the important bit, when it's in this state I can't seem to get it started at all. Sometimes I can get a little rumble for a few seconds but usually nothing.
When you open the fuel cap air there's some positive pressure in the tank. Opening & closing the tank doesn't change anything (I've never had first hand experience with vapour lock, but it seemed like it was worth a test). The weather here's around 15-25C, nothing extreme.
The carburettor is an EL46, no hesitation off idle issues, and from a cold start it's fantastic. Sometimes after a warm restart (or one that's just on the edge of getting it going again) it can be a little reluctant to idle until it gets back up to temperature, and sometimes needs a few blips to get itself sorted out. When I say warm here, if it's running, I stop it to roll a few logs and start it again within a minute or two I'll call it hot, and those starts are fine. Warm would be maybe up to 10 minutes or so, and much more than that I'd be expecting it to get stuck again.
I have been having an ongoing issue with really fine dust getting past the filter. I think it's the crappy bark on these gum trees. I'm pretty confident I'm on top of the problem now that I've got some spray oil for the filter now, and have started running a small line of grease around the outside edge of the filter holder in case it's getting around the filter (I found a thread about that). I took the carburettor off last night and cleaned out some obvious fine dust (that I really hope hasn't done damage to the cylinder), I haven't tried it again yet.
I did come across a comment (that I can't find now), where someone described exactly this problem as one of a few known issues that Husqvarna were working on, but the author had not been active on that forum for quite some time. He mentioned in the brief comment that it was related to a check valve in the carburettor, I think he said impulse line but I can't find the comment right now.
It's possible that this has always happened with this saw, I'm a home owner with a nice saw because I knew eventually I'd get around to the job I'm doing now. Previously I'd only really do 1 tree at a time, or take enough of a break between cutting to not notice. We finally got around to having 20 trees dropped more safely than I could do them, so now it's clean-up time and I'm out there any time I can get away from work.
Possibly related, it does sometimes (not often) seem to get itself flooded on a restart even though I'm not using the choke (just fast idle). When this happens I can get it started as you'd usually start this saw while flooded (throttle + knee + lots of careful). I think this is unrelated but just thought I'd throw it in here.
Can anyone give me some ideas of where to go from here?
Never heard that before. 
