my 2004 rx-8 just began doing this. After I drive for a while, I have to wait about 15 minutes before starting again. The motor cranks but just doesn't seem to catch.

 

doesn't sound good.
How many miles on your engine? When did you last change coils/ plugs? Have you had your cat checked?

 

118,000 miles, plugs are about 18 months old, as is the cat.

 

If the coils and plugs are good, take it out on 100 mile trip with a can of seafoam in the fuel tank, and see if that does it. I was having the same problem and it went away after a week of commuting 50 miles each way (it might have been resolved before then; but I didn't think to test until the end of the week).

 

Im going to put my money on the starter and call it a day. Those with older starters have known and documented issues with the starter. Do a search on warm start issues and I'm sure you will come up with a good number of posts on it along with the model number you need to look for when buying a starter.

 

check the connections on the starter and make sure they all in good condition, because I was having the same problem and I even replace the starter but it was the connection that was loose, after I fixed my problem went away

 

Time for a compression test after you rule out the coils.