After it has been driven and parked for an hour or more the engine will turn over and fire once but not start. After sitting for another hour it starts just fine. Months go by before it happens again. 1987 rx7

 

That is a fairly common hot start problem. The normal cause is simply poor tune (when is the last time the car had a full tuneup?) and/or leaking injectors.

 

Another cause of hot start problems is low compression.

 

The hot start problem you describe is fairly common in the 86-88 cars. The fault is usually blamed on (as stated above) poor compression, and/or leaky fuel injectors. The 89 and later cars implemented a hot start process, such that they cut off the fuel injectors if you engaged the starter with full throttle. That they implemented this feature in the later cars suggests that maybe it isn't always due to low compression or leaky injectors.

My 86 occaisionally has the same issue. Last year I installed a 'hot start' button, essentially a momentary contact switch which, through a relay, cuts off power to the injectors. TriPoint Engineering used to have a kit they offered, but I just built my own from parts at the electronics store. But that is a band-aid fix (though you can also rig it to cut off fuel to the injectors whenever your alarm is tripped, making it that much more difficult to steal the car).

I just completed having my injectors professionally cleaned (RC Fuel Injection, Torrance, CA). Their test report came back indicating that one of my primary injectors was leaking. That would be consistent with the hot start problem. We will see, over the coming summer, if having this done has eliminated the problem. For the time being, I am keeping my hot start switch installed...