Pretty much one of my first official acts as King of Germany would be to ban these awful buzzing boxes. It's not the sound pressure itself that's disturbing, but more the frequencies. Certain frequencies or frequency mixtures simply feel annoying, even without a jet fighter or a pneumatic drill actually exceeding the measured sound pressure.
He said he constantly hears a humming.
What do you think could be done to dampen the sound a bit?
You would first have to find out exactly what bothers him and where. You can actually hear a hum as a hum or a buzzing as a overtone of a hum where you organically hear the buzzing and the brain "recognizes" the hum underlying it. But it can also be that the hum has a frequency that creates a standing wave between your house walls. So it really depends on whether it is better to dampen the sound, scatter it better, or eradicate it at its root (for example, by replacing a wobbling bearing if that's the reason). So you need to find out: is it vibration, is it resonance, is it reflection...
And then after several years the neighbor suddenly notices that the heat pump is too loud? What kind of device do you have?
Yes. If the disturbance (or even just what makes the frequency mixture unpleasant) comes from a worn-out fan bearing or something similar, it can indeed escalate beyond the tolerance threshold only after a few years.