The recommended way

Ubuntu 23.04 has both the packages you seem to need and in the versions that you require i.e. gcc-13 and gcc-13-aarch64-linux-gnu in the [security] [universe] repository and Ubuntu 23.10 has them in the [main] repository ... So, my advice would, naturally, be to upgrade your system to Ubuntu 23.04 or even 23.10 and then install your desired packages like so:

sudo apt update && sudo apt install gcc-13 gcc-13-aarch64-linux-gnu

That is the recommended and safest way.

The hacky way

Otherwise, it's the on your own risk way ... e.g. adding a PPA such as you did ... Yep, you have already chosen the "on your own risk way" ... Read this (emphasis is mine):

Adding this PPA to your system

You can update your system with unsupported packages from this untrusted PPA by adding ppa:ubuntu... to your system's Software Sources.

... that text is quoted from the Launchpad link you included in your question and it's not limited to that specific PPA (which might be of good reputation), but it is what it is.

That PPA enables for installing the package gcc-13 on Ubuntu 22.04 by essentially including that package with its dependencies that can not be satisfied from the Ubuntu 22.04 official repositories and of course some other work might be involved like modifying post-install and/or pre-install scripts and ensuring none of the added package or its dependencies will conflict with existing essential system packages and so forth.

In the case of gcc-13 and gcc-13-aarch64-linux-gnu, which are also made available in Ubuntu 23.04/23.10 official repositories, one might ponder the possibility of installing them from 23.04/23.10 repositories on an Ubuntu 22.04 system? ... Well, yeah surely possible (for these two) and it should make the "on your own risk" zone a bit more appealing as those repositories are official and trusted, but the unsupported part will still apply.

Anyway, I have quickly traced those two packages and their dependencies and then installed both from the official repositories of Ubuntu 23.10 on an Ubuntu 22.04 system successfully without any noticeable drawbacks AFAIK ... After all the GNU C compiler is not essential to the functionality of Ubuntu and is offered as an optional package for manual install.

However, there is probably most likely certainly an extremely big problem awaiting if you're not careful enough i.e. you must make sure no other packages from that repository get installed by any means including automatic-updates so turn all those off and fully update your system first, then follow all instructions precisely (still on your own risk of course) ... If other packages got installed, they might break your system so dangerously badly and possible beyond any applicable repair and of course this is as unsupported as your PPA installed packages ... So, we will not provide any support for either.

Needless to say that I don't prefer it or like it this way, but you seem to need it so I wrote it.

The idea is, basically, to add the official Ubuntu repository containing those two packages, refresh the local cached sources list, install those two packages and promptly delete the added repository from you system like so:

First, add the repository:

echo "deb http://cz.archive.ubuntu.com/ubuntu mantic main" |
sudo tee /etc/apt/sources.list.d/temporary-repository.list

Second, update package lists:

sudo apt update

Third, only install gcc-13 and gcc-13-aarch64-linux-gnu:

sudo apt install gcc-13 gcc-13-aarch64-linux-gnu

Fourth, delete that temporary repository:

sudo rm /etc/apt/sources.list.d/temporary-repository.list

Finally, update your cached packages lists:

sudo apt update

Notice

• You might need to first remove the gcc-13 package you have already installed from that PPA and the PPA itself to avoid possible dependency version mismatch.

• You'll need to run those by version number i.e. gcc-13 -v and aarch64-linux-gnu-gcc-13 -v and not just gcc or you can use the Ubuntu alternatives system to choose the default version.

Maybe aarch64-linux-gnu-gcc is not installed. Try

sudo apt-get install gcc-aarch64-linux-gnu

As of Ubuntu 18.04 you can do:

sudo apt-get install gcc-aarch64-linux-gnu
aarch64-linux-gnu-gcc -mcpu=cortex-a53 hello_world.c

The package gcc-aarch64-linux-gnu is at version 4:7.3.0-3ubuntu2

However, for Raspberry Pi, you should just download the official binaries from https://github.com/raspberrypi/tools which is the more reliable way to do it as explained at: https://raspberrypi.stackexchange.com/questions/64273/installing-raspberry-pi-cross-compiler/83215#83215

Finally, for bare metal, I was not able to find the analogue of arm-none-eabi-gcc, I wonder why: https://github.com/cirosantilli/cirosantilli.github.io/issues/68

You can download a functional toolchain from developer.arm.com and install it manually after removing your existing gcc-arm-none-eabi package.

Go to that website, click the "Download" button and get: gcc-arm-none-eabi-7-2018-q2-update-linux.tar.bz2. Save it in your home directory.

Make sure you've uninstalled the old Ubuntu packages.

sudo apt remove binutils-arm-none-eabi gcc-arm-none-eabi libnewlib-arm-none-eabi

Untar the new package in your home directory:

tar -xjvf gcc-arm-none-eabi-7-2018-q2-update-linux.tar.bz2

Add the new toolchain to your path:

export PATH=$PATH:/home/(your user)/gcc-arm-none-eabi-7-2018-q2-update/bin/

At this point you should have a working ARM compiler and toolchain. (For the Unix newbies: if you close the terminal and open a new one, you'll have to re-run the export PATH statement so the compiler will get picked up again.)

To check if the compiler is installed:

gcc --version