Q: Is it more efficient to keep keep a swimming pool warm or let it get cold and heat it up again?
The one and only gain from leaving a pool heater on is that it will be warm when you get in. You can gain a lot of intuition for how heat flows from place to place by imagining it as a bunch of “heat beads”, randomly skittering through matter. A difference in temperature leads to a net flow of heat energy. So, if the surrounding air is 60°, then an 80° pool will shed heat energy twice as fast as a 70° pool. While the equation and the intuition are straight forward, actually solving the diffusion equation in almost any real world scenario is a huge pain.