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So I like to keep a gallon of water in the fridge so I don't need to make ice cubes, and that I have cold water to drink. Every time I fill up a glass, I can't help but think about how much overall energy is being used and what method to use is the most efficient. Up until the past few days, I have been topping off the gallon after every glass or two, then placing back into the fridge under the logic that it would require less work to pump a smaller amount of heat out than if I were to refill after it reached the bottom. Now, a refrigerator works by pumping heat from the inside of the box to the outside, and the cooler the external temperature is, the more efficient the fridge becomes (for a very basic (and potentially incorrect) description, the coolant is pressurized on the outside of the fridge, it cools to ambient air temperature, and is pumped to the inside of the fridge and is cooled further due to pressure and temperature being inversely correlated. The cooler fluid is then repressurized (heating up), cools to air temperature again, and the cycle continues. When the air temperature is cooler, then the pump doesn't need to run as often, meaning less energy is used). So we recognize that heat should only be added to the fridge when the external air temperature is the coolest (so night time), and it's likely that it doesn't matter whether you spread out cooling of the water over the course of the day, or just do one big cooling in terms of how much thermal energy needs to be removed in that 24 hour period, so I conclude that it is more efficient to do a single refill in the evening before going to bed.

Additional efficiencies that we could account for is how much energy you expend lifting the gallon of water. If it's constantly full, you will epend more energy moving it than if the mass descreases over the course of the day - another point in favor of a single, nightly refill. On the other hand, if we assume power is generated by solar panels, then it is more efficient to fill during the day when the tapwater pump and fridge both are consuming more efficiently generated power.

I am neglecting the number of times the fridge is opened because I am assuming twice per drink (once to take the jug out, once to return it), the same number of drinks will occur for each, and the final refill will also include a drink prior to refilling.

All in all this is a completely pointless use of time, so I would not worry about going into this much detail, because significantly more energy is wasted simply by opening the fridge door, having poor sealing on the fridge, using luke warm water vs cold water from the tap, and thousands of othe variables that make this thought process a silly use of brain power, but hey, we're not talking about efficiency of thoughts, that's a totally different topic.