I had the same problem and I figured out how to fix it! For Samsungs, Apps ‐> ChatGPT -> Battery -> Unrestricted completely fixed the issue for me, it continues thinking/outputting in the background now. Should be a similiar setting for other Android distributions. Basically, it wasn't the app's fault, the OS is just halting it in the background to save battery.

Thank you! That fixed it for me on Pixel 8 as well. Would be great if the app suggested this as a fix.