It could be. Your mower should have an inline fuel shut off valve , so you could shut it off for 10 seconds before switching the ignition off. If it then restarts, you were on the right track. Too bad the valves aren't easier to get to. There is an igniter in the ignition system which can create hot start issues as well.

Thanks tomplum for the suggestion of shutting off the fuel line, indeed it does have a shut off valve. I should have thought about that.

I believe I replaced that igniter several years ago, at that time I wasn't getting any spark at all, the replacement fixed that. I'm pretty sure it's a carb problem even though I always use Stabil with the fuel and at the end of the season I run it dry and empty the fuel tank.

I will try shutting off the fuel with the valve, then try to start it at fast speed. If it starts at fast speed I'll have to remove the carb and buy a rebuild kit. I'll wait for the spring to do that, it's getting too chilly to be working on mowers now.

The mower is a PITA to work on and parts are quite expensive.

"The mower is a PITA to work on"
Second that! Check the air filters and try a new spark plug for giggles.

Looking at the carb diagram (below) I think the part I will need to change is the needle valve, part # 28.
I never worked on a Kawasaki carb before. I'm assuming the needle valve has neoprene rubber at its tip, perhaps that has deteriorated and causing the float to leak gas into the carb. On other carbs I worked on there was also a rubber seal inside the carb housing that needed to be replaced. The Kawasaki carb doesn't show that part.

Hopefully the carb image will appear.

{{gwi:143573}}

That would be the culprit. If you are removing the carb to replace the needle, I would order the bowl and mating gaskets. It isn't necessary to remove the whole assembly if you are steady enough. I've not had a leaking Kaw float, but it would be prudent to check for presents of fuel inside or simply change it. You will want to be certain that the leak is not being caused by debris and in either case you will want to know the fuel system is clean and a fresh fuel filter installed. If you decide to go further to disassemble to clean here are a couple tips. Should you decide to remove any jets or emulsion tube in the stem for cleaning, use the best fit screwdriver and deliberate action to remove them or they can twist the head off. Plus keep them cataloged so they go back in the same holes that they came out of.

Thanks for the tips on the carb removal, should I attempt it won't be until spring.

Is screw 21 at base of bowl for idle adjustment only, can this be a factor why it only starts at slow speed when warm?

Update,
What an idiot I am, screw 21 at the base of the bowl is the bowl fuel drainage, please disregard my earlier post.

I did try the method of shutting off the fuel supply for 10 seconds before turning iginition off. It still behaved the same way, only starts on slow speed.

I may just live with it since I have for years. I don't want to risk messing with the carb since there's a chance something else can go wrong as indicated below.

I recently changed the mower belt, not hard to do. While I was at it I thought I'd change the primary belt which was still in good condition. Big mistake changing the primary belt. After I finished I was checking it out while still jacked up, no matter what I did as soon as I put it in gear it stalled. After spending countless hours rechecking my work I gave up.

I lowered the mower from the jack and was prepared to push the 357 pound machine into the shed. Before doing so I started the engine and put it in gear, everything worked again. It appears by having it jacked up caused the problem. Moral of the experience, if it ain't broke don't fix it.

I put traction belts on these a few times. One time I wanted to pull my hair out. The belt was missing, of course not lots of room, but it didn't seem horrible. That was until I realized that I had the routing off, and scratched my head and knuckles for an extra hour figuring it out! 3 pulleys. How hard can it be? Further insult to injury came about 1 week later and someone out of the blue handed me a service manual for one of these he found and thought maybe I could use one! Fortunately there aren't lots of these around anymore...

I feel a little better now, so I'm not the only one having a hard time changing the traction belts. At least you had an excuse, no belt present and no manual.
My belt was on and I have the owners manual which has an illustration of the belts. I suppose growing older my senior moments are more frequent.

From now on before taking anything apart, I will take close up pictures with my digital camera at each stage of taking things apart. Sometimes while waiting for a part, you can easily forget how things were assembled. A good color picture can be very helpful, better then diagrams in a manual.