You most probably have the very common "hot-start" problem which many RX-8's have or had.
To find out what might be causing it more info is needed such as;
What's the mileage on your ignition? (near or above 25,000 miles?)
What starter motor do you have? (the serial number is on top of the starter)
Was your RX-8 flooded? (if so your plugs are probably fouled)
Also note that the stock coils, sparkplugwires and sparkplugs unfortunately don't last very long and the starter motor is very weak from factory;
The RX8 Ignition coils are prone to fail, so it’s probably a good idea to change them.
You can also check the epoxy fill on the bottom for burn spots and /or surface flaws. It is usually smart to replace them and start fresh.
Relocating the coils to improve cooling is another possible but more difficult option that should improve life expectancy.
The stock wires have a rubber coating which cannot handle heat for a very long time before they start to crack which let's moisture in and causes interference, replacing them could be neccesary. If you have more cash to spend go for a set of silicone wires which will fix the heat influence problem.
Your sparkplugs might need replacement too on higher mileage, especially if the engine was flooded a few times which basically ruins your plugs.
Make sure you get 2 Leading (RE7C-L) and 2 Trailing (RE9B-T) spark plugs and put them in the right locations or you might cause serious engine damage! Don't be shocked of quite high prices as the spark plugs are a special type for Rotary engines (no alternatives here).
If you make hot starts very often you might want to switch your Leading spark plugs for the RE6C-L type which Mazda introduced in 2006 to combat the hot start problems.
As for the starter engine N3H1 is an old 1,3kW starter which doesn't work very well, especially when old. N3H1-A is an upgraded starter and the number N3Z1 means you have a 1,6kW starter later introduced by Mazda. If you have extra cash to spend you could also go for a 2,0kW upgrade.
I hope this gives you some useful information, and I wish you luck!
Also if you eventually plan on replacing your complete ignition in a while you could buy mine for a fair price as I am making an early upgrade within a few months from now. Of course all parts will be checked upon first to make sure they are 100% ok before they are sold.