I don't see how that is possible. Either way you need the same amount of heat added to the tub. Maybe you save a little because the return pipes are not cooling the water, but otherwise the needed heat should be the same. I personally insulated all my hot water pipes which solves that worry.
Though fresh water and letting out the old will clean the water you are soaking in, which may be valuable.
The way ecocute works is that it heats water at times when electricity is cheap and stores it in a highly insulated tank. So if you’re having a soak in the evening, it’s cheaper to use the water that is already preheated at cheaper rates and stored, than to reheat it again using the expensive to peak time rates.
My rates are 36yen from 6am to 10pm and 19yen on the other end.
Re-reading my earlier post, I should’ve said cost and not energy.