The brief
This S5 had a coolant level that kept going missing, dropping between checks with no obvious puddle, so it came in to have the leak found. Coolant that disappears is going somewhere, and a slow leak turns into an overheat the day it lets go, which can take the engine with it, so it needed pinning down. A common coolant leak on these is the water pump. The pump circulates coolant around the engine, and it seals against the block with a gasket and runs on a shaft seal, both of which wear over the years until the pump weeps coolant from around its housing or the shaft. It often weeps just enough to drop the level slowly without making a puddle, the engine heat drying it off. A weeping pump doesn't reseal, and the thermostat lives in the same module on these, so the two get changed together.
The diagnosis
A pressure test on the cooling system pinpointed it, the water pump was weeping coolant from its housing and shaft, which is the slowly disappearing level. The radiator, the hoses, the expansion tank and the rest of the system held fine, and the thermostat in the same module was due alongside the pump. That's a water pump and thermostat module replacement, plus a bleed and pressure test, rather than chasing a weep that only gets worse.
The work
The cooling system was drained, the old water pump and thermostat module removed, and a new genuine Audi-spec pump and thermostat module fitted with a fresh seal, the drive that runs it checked and refitted to spec. The system was refilled with the correct Audi coolant, bled the proper way so no air pockets were left, and pressure tested again to confirm it held with no weep. A road test confirmed the gauge sat steady, warmed up at the right rate, and the level stayed put.
The outcome
No more coolant loss, the level holding between checks, the gauge steady, the engine warming up at the right rate, and the system holding pressure. The S5 went home with the leak resolved. A weeping water pump only gets worse and ends in a sudden coolant loss and an overheat, so replacing the pump and thermostat together kept it to a tidy, planned job.