My own mk5 rs2k had this problem, and for me the solution was a thickish black/red wire that is the switched live to the starter motor solenoid. There is a connector near the battery that becomes corroded over time and limits the amount of voltage going across the connection, and you need max volts/amps when trying to start a hot engine!

I cutout said connection and soldered the 2 wires together, never had the problem again.

I used to have to choose petrol station that had a slope so that I could bump start the car after filling up! But I was once not able to find one, so had to try and cool things down by pouring a 2 litre bottle of water over the starter, then re-filling it and repeating, after about 20 mins the attendant came over and said I must have a very bad leak in my cooling system if I had nit yet been able to fill it up by now! Anyways I roped him in to giving me a push start