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'80 528i - Cold starts just fine. When temp goes over 90, starting becomes intermittent. No crank, just a click and a hum (potentially from fuel tank). Battery is charged, alternator newish. Usually will start up after 30-60min of "cooling off".

Could it be the starter?

Pressure in the fuel tank?

Coolant temp sensor?

Any advice would be golden. Thanks from Iowa!

If the engine doesn't turn over, probably the starter is at fault.
Can you hear the starter whine when the problem occurs?
Had a similar problem on mine, got rid of it temporarily by adding a relay to the the starter solenoid supply so that it doesn't get voltage through the ignition contact at start (which is another problem on my car). Fed by a relay directly from the battery, the starter now couples all the time. The bit of extra voltage seems to help things just enough to push the pinion also when hot.
In my case, probably things down there in the starter need some cleaning and greasing.

You can test this hypothesis by trying to couple the starter solenoid to the battery +12V with a thin (thin means short circuit protection) wire next time you encounter the issue. Be careful with electrics though, you don't want to make a short circuit to any of the metal parts (engine etc) with a wire connected directly to the battery.

Thanks Falkenberg - luckily (however not conducive to troubleshooting) we've run into a little cooler weather here, I've narrowed my focus on the starter, as I can here it hum when I turn the key.

G. Fischer wrote: I've narrowed my focus on the starter, as I can here it hum when I turn the key.

Sound like the same issue as on mine, the pinion is not pushed out when hot.