Ignition start problem

[Audimike008]

Hi guys been having trouble lately with my 2008 a3 170tdi seems to be ok starting from cold start but if been on a journey ie stop to get fuel it struggles to start again and will only start by rolling it has anyone had this problem ir no anything about it ? My guess is the starter motor

 

[LEE69]

Coolant temperature sensors do this if faulty, they send the wrong signal to the ecu and then it can't inject the right fuel/air mix.
If you can get it scanned, i think a fault in the cooling system would show.