Alternative OS and battery usage. What alternative OS is the most efficient?

At the moment I use a FP3+ on Android 11 on the KPN network in The Netherlands. The update (or more correct: upgrade) to Android 13 seems to be still delayed for FP3(+) owners/users on KPN. I am not sure if the update (or upgrade) to Android 13 wil be canceled. Therefore I consider installing an alternative OS on my FP3+. One of my main criteria choosing an alternative OS is battery usage. My question is: does using E/OS, Lineage OS, Ubuntu or something else that is not Android on a FP3+ draining the battery less than Android 11? If so, what alternative OS is the most efficient?

I haven’t done any direct comparison by myself, but I suppose any custom ROM that’ll give you more control over background services will be easier on your battery in-between active usage periods. Google services phoning home in the background are not to be underestimated when it comes to this.
I’m not sure wether there’s the one ‘most efficient’ OS. Just look into some custom ROMs you like. LineageOS comes without Google apps out of the box for example. Or choose something like CalyxOS that uses the open source microG implementation as a replacement for core Google services. Just be aware that despite best-efforts not everything depending on Google stuff will work perfectly.