There are two great apps for this I’ve found for Android: DreamLab and BOINC. Android devices are great because they are some of the most efficient-per-watt computation available, and cost anywhere from 25 cents to a few dollars to run per year depending on your device and power cost.
BOINC is a network of many different projects, you can pick and choose which you want, just install the BOINC app and attach to the projects you want. It’s a tad more complicated than DreamLab but way more powerful. There’s medical research (World Community Grid), space research (Einstein@home, Asteroids@home, Milkyway@home), math research, and more. Note that the version in the play store is basically non-functional, you’ll need to download from F-droid.
Note that many Android phones are not designed to effectively vent heat from 100% CPU usage constantly. Both of these apps allow you some degree of control over CPU usage, I generally set my devices to 50%. To be safe, remove batteries while these devices are running. It will help preserve battery life, increase power usage efficiency, and important will prevent your battery from becoming a dangerous swollen fire pillow.
Please also be mindful. These old devices don’t receive security patches and are going to be connected to the internet. Any exploit or bug and you’ll have a compromised device with local network access and even gps in the hands of bad guys.
There’s also the battery as OP said. It’s better if you take it out, if possible. Do not leave it running on 100% all the time or you’ll have a fire in your house!
The CPU will automatically scale down in most devices to keep the temp in a safe range. Batteries will swell for a while before they become an issue. As long as you aren’t browsing the open internet with it and are behind a NAT, patching probably isn’t a huge issue. But of course, always be mindful of how you use your devices particularly when it comes to security. A factory reset will remove all sensitive data from the device and give it a computational power boost as well.
Re: heat management. Putting the device on a rack (like one used for cooling cookies) and removing the case and/or cover can make a surprising difference. Heatsinks can also be applied to the case itself or to the processors themselves. I have several devices running at 100% with no issues thanks to some small heatsinks repurposed from a raspberry pi. You can get like 100 of them for $10 on amazon/ebay/etc.
It was late yesterday and I didn’t realize I made it sound scary! Yes to all you said! The heatsink is a great idea, too.