The heat is conserved. If the waste water is hotter than the incoming water it will conduct on its own (and if they're arranged in back flow the equilibrium wil

The heat is conserved. If the waste water is hotter than the incoming water it will conduct on its own (and if they're arranged in back flow the equilibrium will be pretty close). Hurrying it up with a heat pump only makes it happen faster and adds the energy you put into the heat pump. You could cool the waste below ambient, but that's no different to cooling anything else outside. For simplicity, if your hot water system was already a heat pump, you could run it past/into the cold side. No real efficiency gain vs exchanging it with the cold water (actually a loss in theory) but might be cheaper.