Rather than beginning with an explanation of why this happens (which would require the reader to have faith in the credibility of a stranger who says things they've never heard of before) I will instead suggest a simple test you can do yourself:
The next time you need to start the car when it has been less than a half hour since it last ran; try using the following technique: Turn the ignition switch on, but stop and wait when the key reaches the position where the dashboard warning lights come on. Wait for 10 full seconds with the key in that position; and then turn the key the rest of the way to start the engine. If it now starts quickly, you may be interested in reading the long explanation that follows.
This problem is the result of a combination of many factors. The primary factor is that Federal and California emission regulations for late model automobiles are so demanding that vehicle manufacturers have been forced to calibrate their air/fuel mixture settings to the leanest possible ratios. Lean mixtures contain less fuel and more air than rich mixtures. The lean mixtures create less pollution, and produce more fuel economy. But lean air/fuel mixtures are inherently difficult to ignite during starting. This problem has become even more severe in the last few years; when additional Federal and California emission regulations required oil companies to add increasing amounts of oxygenating additives to the fuel they manufactured. Oxygenating additives further reduce vehicle pollution; but they make starting even more difficult, and they also create significant amounts of carbon buildup on intake valve stems (which is a whole new problem we now face).
The second factor is that today's fuel injection systems are equipped with temperature compensating features which add additional fuel to richen the mixture when the engine is cooler than normal operating temperature. This is done because cool or cold engines will not run or start well with the lean fuel mixtures that are used at full operating temperatures. So, as the engine is shut off, and then cools down from full operating temperature; after it cools for 30 minutes or longer, more fuel will be added to richen the mixture during starting; which in turn makes it easier to start. On the Mazda, it takes 30 minutes for the engine to cool down to the point where the mixture becomes rich enough that it will start more easily.
The third factor is that the fuel injection system in modern cars is supplied with fuel by an electric fuel pump. This pump is typically located inside the fuel tank, or underneath the car next to the fuel tank. Because of the distance between the fuel pump (which is at the rear of the car) and the engine (which is in the front of the car), the fuel has to pass through a long hose in order to reach the engine. When the engine is shut down; fuel will drain back into the gas tank, leaving the fuel hoses partly empty. When the engine is restarted, the fuel pump has to refill the hoses with fuel, before enough pressure can develop to allow fuel to flow into the engine at a normal rate. It takes about ten seconds for the pump to refill the hoses with fuel.
If the engine is started before the fuel hoses are filled, the initial lack of fuel will make it far more difficult to get it running. In some engine designs, when the engine is cold, the extra richness added by the temperature compensating circuits will override this situation. Other vehicle types will start easily when the engine is hot; but become difficult to start when cold, unless you then wait 10 seconds with the key on. It all depends on the engine design and the calibrations of the fuel injection system.
The fourth factor is that most drivers are not aware of this situation. This is further compounded by the negative PR that would be created if vehicle manufacturers told people who bought their cars that it is necessary for them to wait for ten seconds with the key on before their car will start properly. Many people would not buy a car if they heard that. So the industry will not talk about this issue. And those drivers who sense that there is a problem are mystified.
To be fair; some engine designs are much more sensitive to this problem than others. But I've seen it in many different makes and models of cars. Those of use who live in California, and use California specification reformulated fuel, have the worst problems of this sort.
I hope this clarifies the issue!!!