Remembering why I
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AO Silver Member
Join Date: Sep 2001
Had pretty much what you describe on another engine once - the ECU water temp sender was OK when cold, OK when hot...but at about 60 degrees it went open circuit, which made the ECU think it had suddenly got rather chilly..... You could drive through it but if you had to stop and idle the car, it would stall.
I know for sure it was this - I had an aftermarket ECU on it at the time and could log what was going on - temp rose gradually as normal, then fell to -50, then jumped back to 70 a few minutes later.
So, yes, try changing it first for sure. It's a stock Bosch item so shouldn't be uncommon.