Unfortunately, there is no PeaZip package in the default Ubuntu repositories, but there are several ways to install it. Here I will describe the two methods that I think are the easiest and better in my opinion.
Method 1: Install PeaZip using the prebuilt .deb
package
The benefit of this method is that it is a straightforward way to install PeaZip, if you are familiar with software installation in Ubuntu/Linux. The downside is that you'll have to manually repeat the installation procedure to install a new version of PeaZip.
Go to the PeaZip webpage and click PeaZip for Linux.
Depending on the desktop environment you use (or your personal preference), click either the DEB GTK2 link (suggested for GTK-based environments, such as GNOME, XFCE, Cinnamon, Unity, etc.) or the DEB Qt5 link (suggested for Qt-based environments, such as KDE, LXQt, etc.) and download the .deb
package of your choice to the location of your choice.

Either double-click the downloaded .deb
file and install the package using Ubuntu software or open a terminal, cd
to the location where you downloaded the .deb
package, and run:
sudo apt install /path/to/peazip.deb
Make sure to replace /path/to/peazip.deb
with the actual path and filename.
To uninstall it along with its configuration files you can use the following command:
sudo apt purge peazip
Method 2: Install PeaZip using Flatpak
The downside of this method is that you have to go through some additional procedure to install Flatpak (if you haven't already). The benefit is that the package can be easily updated to a newer version by running in a terminal flatpak update
(which by the way will also update all your other Flatpak apps) and you will also get access to the Flathub repository (which is great in my opinion).
Install Flatpak by following the official instructions for Ubuntu.
Install PeaZip by running the following command:
flatpak install flathub io.github.peazip.PeaZip
To uninstall it you can use the command:
flatpak uninstall io.github.peazip.PeaZip