Hot Tub heating connundrum 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!
 A: Given reasonable insulation, cover closed, etc, the rate of energy loss is proportional to how much hotter the water is than ambient.  That means more energy is lost from the water if it is kept continuously at high temperature.
However, the total energy used has to look at the overall system.  It does take a large amount of one-time energy to get the water from ambient to service temperature.  The power to do that in a reasonable time is higher than the power to maintain the water at service temperature.
The efficiency of whatever system is delivering the energy to the water could vary with the power.  Providing the large initial slug of energy to bring the water up to temperature could, in theory, be less efficient to the point of overall energy use being less if the water is kept at a steady temperature.
However, that's a highly theoretical argument.  You didn't say how the water was heated, but if it's directly from electricity by resistive heating, then the conversion from electric to thermal energy is pretty much 100% either way.  Other systems might even be more efficient at higher power.
I point out the high power inefficiency mostly as something for you to think about since it appears you are overlooking it completely.  In reality, I would be very surprised to find a system where keeping the water hot all the time uses less energy long term.
Another point to consider is the cost of the energy.  That can certainly vary significantly by power and by time of day in real world circumstances.  In some places, the cost of electric power is much lower during certain hours of the night, for example.  In such cases, it might well be less costly (not necessarily less efficient) to bring the water to full temperature every time just before the low cost period ends.  The water would gradually cool during the day, then be heated to full temperature again next night.  It's no stretch at all that a real world system like this could be cheaper to operate than one that lets the temperature coast down, then brings it up to full right before use during peak rates.
Of course, hot tubs aren't just about energy usage or cost.  They are a luxury item, which by definition means cost is not a major driver.  To serve their purpose, in some usage cases, they have to be ready for use without hours of advance notice to get the water to operating temperature.  That is ultimately what drives when a hot tub is heated and how hot it is kept.  Nonsense physics arguments to make someone feel better about all the wasted energy can't be countered by Real Physics, since it's not about physics in the first place.
A: Thanks everyone for your answers.
After reading through, I realise this can be a lot more complicated than first meets the eye.
However, putting aside various points such as cost of electricity at different times, and the need to have the tub available for immediate use, there is one incontrovertable point which is that the hotter you keep your hot tub the greater the (energy)cost because the greater the energy loss. 
Going back to my point about trying to explain to the 'lay-person' about a subject like this in terms of physics is like trying to convince them of a political or religeous belief - not worth even trying! The particular 'belief' I have now encoutered twice is 'it only takes a little power to keep something hot continuously but a massive amount of power to heat something from cold'.
Interesting point from 'sammy gerbil' that this is very similar to heating your house, because the person I was speaking to the other day about her hot tub backed up her point by saying that her son-in-law came to stay and put the heating (in their house) on hot continuously during his stay, and she reckoned they used less heating-oil (confirming belief of said above point).
Conclusion: Like people who come out with ridiculous conspiracy theories better to walk away than try to convince them otherwise - you get angry and they think you're saying they are thick - not likely to end well.
A: I'd like to think about the principal of conservation of energy in this matter.
I will use two extreme approach.
First, let's assume that the environment is ideal, it's in vacuum environment, 100% insulated, so zero energy loss. In this case, we only need to use the energy once to heat up the water and the water will stay at the same temperature forever.
Second, let's assume that the environment is very cold and insulation is very poor, and with 2K watt heater, no matter how long you turn it on, the water will only reach temperature of 35 degrees celcius (reach equilibrium) because of so much heat dissipation to the surrounding.
On the first case, it doesn't matter because you only need to turn on the heater once and there is no debate about turning the system on stand by or turn it on once shut it off forever.
But on the second scenario, very obvious that it will use less energy if we only turn on the heater when we need to have 35 degrees celcius of water.
How about in between? The same. It will use less energy if we only turn on the heater when we need it.
The "additional" energy only needed if there is dissipation of heat from the system. So the total energy "wasted" in this process is equivalent to the amount of energy dissipated. And the total energy dissipated is greater if the temperature difference is greater, meaning 40 degrees celcius water will dissipate more energy in 10 degrees environment as compared to 15 degrees water.
But why all the hot tub companies recommend to keep the hot tub hot all the time? Because it's for convenience, it's not about physics or energy saving.
