Could be the car is running a bit fuel rich . ie a bit too much petrol in the fuel/air ratio. Cold starts need a richer fuel mixture and so it starts fine.(In the 'old' days you used some choke). Once the engine is hot the ECU should reduce the proportion of petrol making the fuel mix leaner. But if its still too rich a recently hot engine may not start easily, or become 'flooded' with too much petrol and not start at all.
There could various reasons for it not starting. It not necessarily true that the recovery service cannot diagnose the problem because the car now starts. They can read if there are any fault codes that may help identify the cause. But they may not have the time or a suitable reader to do so.
Sometimes an actual faulty component such as fuel injector, spark plug etc causes the problem. Other other times it may be a faulty sensor giving the ECU incorrect data which it attempts to correct by changing settings, but actually making things worse. Things like air flow, fuel flow, temperature, CO2 level in exhaust gases etc.
I am assuming the car has been recently serviced and is not overheating due to a shortage of coolant etc. Its tempting to start changing sensors ,replacing parts , cleaning parts etc in the hope of stumbling upon a cure. But the cost can add up. It is better to get the ECU read first which may immediately identify the problem.
