They say a sponge has an estimated effective lifespan of two months, so I don’t think hard water should cause issues over that timespan.
Since the brushing process happens at a roughly constant temperature, this is going to behave more like a sink than any appliance that heats cold water and puts out hot.
I wouldn't worry about it if I were to try this. The fact that sponges turn gross in short order would be much higher up my list of reasons to be concerned.