The brief
The 530i had been overheating, the temperature climbing higher than it should and the engine running hot. The owner stopped driving it and brought it in, which is the right call, an engine that overheats shouldn't be run on. Overheating often traces back to the thermostat. The thermostat is the valve that regulates how much coolant flows to the radiator. It stays shut while the engine warms up, then opens to let the radiator do its job. When it sticks closed, the coolant can't circulate properly, so the engine overheats fast, and even when it sticks open the engine can run hot under sustained load while taking too long to warm up. Either way, a thermostat that's stopped regulating cleanly needs changing, and an engine that overheats can warp a head.
The diagnosis
A check of the cooling system traced the overheating to the thermostat, which wasn't opening cleanly at its rated temperature, so the coolant wasn't circulating to the radiator as it should. The water pump, the radiator, the hoses and the level all checked out, so the thermostat was the fault. That's a replacement, with a fresh seal, rather than trying to free up a sticky thermostat that's only going to stick again.
The work
Enough coolant was drained to reach the thermostat, the old unit removed, and a new genuine BMW-spec thermostat fitted with a fresh seal. The system was refilled with the correct coolant, the air bled out the proper way so no pockets were left, and held under pressure to confirm the seals were dry. A road test confirmed the gauge sat steady and the engine warmed up on time.
The outcome
Gauge steady through traffic and at speed, the engine warming up on time, and no overheating. The 530i went home with the cooling system regulating properly again. A sticking thermostat only gets worse, and the failure at the end is an overheat that can cost a head gasket, so changing it kept it to a tidy, planned job.