I’ve experienced with bike parts sourcing that Google search doesn’t always go deep in online stores catalogs, so I usually search (or navigate) on each store website itself.
More to the topic, I expect Purism to sell Librem 5 batteries as well, but maybe not before Evergreen shipping starts or ramps up.
I will admit to sneaking a peek at the PinePhone May Update site to see where they are. Overall, remarkably similar progress compared to the Librem 5, overcoming many of the same obstacles, including suffering trolls saying “…all the vague commitments regarding shipping. And when a company does not answer legitimate emails…” – classic!
This part jumped off the page:
“…a-wai, the developer behind the popular Debian+Phosh build, has christened the project Mobian – the recent Mobian build implements crust deep-sleep allowing the PinePhone to run for upwards of 12 hours with radio (WiFi and LTE) connectivity enabled…”
Since Purism is based on Debian and Mobian takes Phosh from Purism, it seems to me that this Mobian innovation could be helpful for the Librem 5. I mean, why re-invent the wheel? 
Maybe, but both radios would be completely different in the PinePhone as compared with the corresponding radios in the Librem 5.
That had occurred to me, but even if the code cannot be taken verbatim, the technique may be applicable.
I’ve made this joke before…
… but it seems in the future the battery could indeed get better, last longer: a new, better variant of Li-S, which also"… would also be cheaper to produce, have a smaller environmental footprint than those currently on the market, and lead to less hazardous waste, the researchers say."
So, I guess we’ll have to stay tuned and ask for an upgraded battery v2 or v3 (also a built in Qi maybe?) in a couple of years.
Purism says that it will sell extra batteries.
The power management improvements in the Mobian are mostly based on Crust, which is firmware for the Allwinner A64 and H5 (with partial support for the A83 and H6):
Crust improves battery life and thermal performance by implementing a deep sleep state. During deep sleep, the CPU cores, the DRAM controller, and most onboard peripherals are powered down, reducing power consumption by 80% or more compared to an idle device. On boards without a PMIC, Crust is also responsible for orderly power-off and power-on of the device.
For this to work, Crust runs outside the main CPU and DRAM, on a dedicated always-on microprocessor called a System Control Processor (SCP). Crust is designed to run on a specific SCP implementation, Allwinner’s AR100.
Crust has been development since October 2017 and still has problems waking up fast enough to take phone calls.
Purism is currently not focusing on suspend to RAM for the i.MX 8M Quad, but instead working on other power management improvements.
… and/or recharge faster.
As not sure if this USB-C power bank with graphene core might significantly extend practical outdoor usage of Librem 5, so asking what is your opinion about it? Anyway, it should be quite usable (instead of carrying around four AA batteries 
, just comparing two sizes) as portable and ‘rapid recharging’ (1:45hrs) 5000mAh power bank.
Besides, here is an answer to @ballgum previous question:
Nice idea but it’s not the internal battery. I’d rather change the internal battery than tether a brick to it. And the size is inconvenient - I’d rather it be double wide and half as thick (or less) as then it would nut bulge and could potentially even be fixed on the back of the phone (or built in case).
Thanks for your opinion! Therefore and although price was tempting I decided to wait (be on safe side) for something like this (bigger one with the same capacity) graphene-composite battery (with pass-through charging ability and USB-A output of 12V/1.5A).
You maybe could/should copy/move case discussion to one of the case threads, but about that extra battery, to avoid thick bulkiness I’d look for tail/end mounted (no cord) batteries too. They’re about as wide and thick as the phone so mostly add only length, adding maybe 3000mAh - a compromise for sure.
At one point, there was some serious discussion in the pc world about potentially using butane powered mini fuel cells to power laptops. The idea is that a laptop that would typically last for only two hours on a lithium battery charge would run on a fuel cell battery of the same physical size for a few days of continuous use. If it gets hot (within reason) that’s okay because there is plenty of extra power available. So wasting current is no big deal. Just refill with butane as needed every few days or maybe once per week. Maybe the same thing could be done with a phone battery.
I really doubt that you can find a generic battery to fit the Librem5 since it has a very odd shape, but the dimensions are listed in the FAQ.
I actually tried to find one and came surprisingly close. Very little effort would be needed to fit a battery from one or two other devices to the same battery slot (no, I will not say which - I haven’t even tested them and they might be hazardous). Not high density though. Just an option to Purism sold batteries, if there will be limitations in availability or pricing. But more interesting should be that, as the back cover is removable (and hence, changeable and modifiable), different battery sizes and shapes should be possible. And Purism might want to create and sell more advanced batteries in the future.
I have now received( October the 3rd ) and tested this master piece of a phone, and I do like the way it works, however the battery life( 4500 mAh ) is in my experience 4 hours 45 mins( 3 phone calls and 7 text messages on average latter 3 days. ) At least my iPhones have like 15-18 hours of battery life in which is It’s like having a Raspberry Pi with built-in UPS for now. If the phone runs 18 hours a day with some activity and had Telegram Messenger and email I would be ready to place this phone in a project.
This is simply my experience this far. Are there other battery options for this? For example I would not mind having a battery that doubled the thickness of the phone.
I tried to put display light down to 20% however this did not improve battery life noticably.
I would love to have a phone that is linux without java between me and linux. Android was never an option for me, its lagging touch system makes me nervous. I do see the potential in Purism and Fuchsia.
With wifi HKS engaged most of the time you should get about twice the battery life easily, I’ve also turned of pureos-store’s package fetching. I usually grab a a small powerbank when leaving the house for longer periods of time. As you don’t mind doubling the phone thickness: I’m using a Mazer EnerG Duo 10C (https://shopee.sg/Mazer-EnerG-Duo-10C-10000mAh-Powerbank-i.74421134.1944552297)
I already have a 30 000 mAh, looking or less like the one you got. It would be nice to have this extra battery in one piece, not as some sort of trailer.
My Librem 5 no longer takes power when charger is connected, the red light comes on, however never changes color or let me turn the phone on anymore.
The red light goes out (doesn’t change colour) when charging finishes normally.
Have you tried booting with all kill switches in the kill position and with the battery removed and with the charger supplying power?
Otherwise probably best to support@puri.sm for various troubleshooting steps.
Ideally you would fault isolate the charger i.e. make sure it is working.
Can it charge anything else (where it can supply enough power at all for the device)?
Do you have a USB power meter? (unlikely but …)
Do you have an alternative USB-C PD charger to try?