Note that most distributions do not feature "app stores" per se, but rather package managers, which are very different concepts. Ubuntu sources most of its packages from Debian (the Linux distribution it is based on). There's already an intent to package Godot in Debian and Fedora, but progress has been scarce on the issue.
You may be more interested in Godot distributed as an AppImage, which is a distribution-independent application bundle, akin to .app directories on macOS. An AppImage compiled on an old distribution can run on most distributions (new and old) and doesn't require any non-default system dependencies to run; thus, you just need to make it executable then run it.
