How to install Gnome 3.38 in Ubuntu 20.04?

No.

LTS releases are intended to be stable in the sense that the software does not change. The software on 20.04, with limited exceptions, won't be updated. That's the whole point of an LTS.

• LTS releases continue to receive security updates from the Ubuntu Security Team, and major bugfixes. These security updates include newer versions of web browsers. However, a whole new desktop stack is neither.

Let's take a look at an example: The Gnome environment from 18.04...

gnome-shell | 3.28.1-0ubuntu2                   | bionic
gnome-shell | 3.28.3+git20190124-0ubuntu18.04.2 | bionic-security 
gnome-shell | 3.28.4-0ubuntu18.04.3             | bionic-updates
gnome-shell | 3.36.1-5ubuntu1                   | focal   

...see how gnome-shell 3.28 was in Ubuntu 18.04? You can see the minor updates from security patches and bugfixes, but it's still on version 3.28. Meanwhile, 20.04 uses Gnome 3.36. And two years from now, it's likely to still be using 3.36.

In general, if you want newer software, you should run a newer release of Ubuntu.