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In addition to things like heat exchangers or reducing the amount of time/water in a shower, is there any work being done around optimizing the amount of time a hot water heater keeps the water at high temp? I’ve thought about this for extended absences but this could also bring potential benefits if it just doesn’t run for certain portions of the day or middle of the night.

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I'm not sure if things are different outside Australia, but it's probably been 15-20 years since I've lived in a house with a hot-water storage tank. As far as I know most places seem to have instant/continuous systems that heat the water on demand as it passes through the line.

Solar-heating definitely makes storage make more sense, but I don't know how the numbers actually work out on that one.

Or for heat pump water heaters, which have unbeatable efficiency. Still not cost-effective in many areas compared to gas-burning heaters, but they use much less energy.

People have done that (some recent post on HN was about using a RaspberryPi to do just that, including a "get hot enough to remove legionaries' disease").

Likely for most people the tankless heaters would be a better option, or using in-floor water-based heating instead. Something like this: https://www.radiantcompany.com/system/opensystem/

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