There are a number of facets to this problem.
Fuel evaporation is just one of them but is quite common.
Next on the list is the earthing of the starter motor - if the battery "feels" flat when the engine is hot and won't turn the engine it is worth getting the earthing and main power feeds checked (try looking up the brown wire mod for some help along these lines).
If the earths are good but the engine just won't turn then the odds are you have one of the engine blocks that are slightly distorted (usually the result of overheating earlier in the car's life). Sadly the only option is a different engine block as the main shell bearings are no longer in alignment when the engine is hot. It saps power when it is running too so could actually save money with a fix in the long run due to better fuel economy.
If the engine turns but won't catch then you most likely have a fuelling or ignition problem. The electric fuel pump and electronic ignition mods will go a long way to help. Swapping the original air filter housing for a smaller chrome or plastic filter/housing goes a long, long way to helping - as does opening up some of the slats in the engine cover (the inner side of the cover and not directly over the block). Again you can find more information on this if you do a search.
The final significant possibility is a loss of compression while the engine is hot. If it does turn freely it is worth getting a compression test done as a precaution. It is not unusual for the 2 and 3 cylinders to cross-leak if there is any distortion in the head but in the early stages it only shows up while the engine is properly hot and that normally happens only after the engine is shutdown. Later engines have additional headbolts to address the issue at least in part.
On this last subject it is worth noting that later engines used stretch bolts while earlier engines did not - this was not widely publicised by Fiat so as a result the stretch bolts have commonly been reused. You don't see it so much in X1/9s but Unos and Puntos were terrible for this and created a bad name for the cars for head gaskets blowing - nothing to do with the gaskets though. I only found out about this a few years ago from a dealer parts manager - there used to be (and may still be) quite a stockpile of NOS headbolt sets. Even some dealerships were ignorant of the fact! The technical information published by Fiat at the time was quite hit-or-miss, the worst case being incorrect torque settings listed for some of the FIRE heads...
