I registered to ask this specific question because it completely does my head in.
I am not so much asking this question because I don't understand the answer, but because I find it massively frustrating not being able to convince people of the physics of it!
I have twice had a chat to friends who own hot tubs about heating the water. Both times they say that it is much more energy efficient to keep the tub hot all the time - this is the line all hot-tub companies put out there because nobody wants to think they will need to plan when to use it and hence when to turn the heating on - they just want to know they can jump into it when they feel like it.
I point out that the hotter the water is, the greater temperature differential and therefore cost in heating it. The answer is invariably that to heat the tub from cold takes a massive amount of electricity, therefore it is more efficient to keep it hot all the time. Also they say that it is really well insulated.
I always wish I'd never brought the subject up as the conversation ends with them not understanding how I can't agree with their point of view (ie how can you not understand the massive amount of electricity needed to heat the tub from cold?!)!
What I really would love to see is a graph of temperature of the water against time, including total energy used for different scenarios (mainly of keeping the water hot continuously, and heating it up to hot say once a day). I have been googling for answers for but there is nothing specifically about this, and you have to try to cut through all the stuff the hot tub companies are putting out there - they just want to make things sound simple and convenient.
This is a similar question to filling up the kettle each time you want to make a cup of tea - I think this is easier to understand as being a waste of energy, although my wife did say earlier "yes but the water is still warm next time you want to heat it up so it doesn't take as much energy" - makes me so mad!