How does one install pygobject from the official repository?

Development libraries usually don't show up very well in the software center. Usually using apt-cache search will help here. So for example:

apt-cache search python gobject

Will show you all the packages related to python and gobject introspection. Here is a snippet of the results:

python-gi - Python 2.x bindings for gobject-introspection libraries
python-gi-cairo - Python Cairo bindings for the GObject library
python-gi-dbg - Python bindings for the GObject library (debug extension)
python-gi-dev - development headers for GObject Python bindings
python-gobject - Python 2.x bindings for GObject - transitional package
python-gobject-2 - deprecated static Python bindings for the GObject library
python-gobject-2-dbg - deprecated static Python bindings for the GObject library (debug extension)
python-gobject-2-dev - development headers for the static GObject Python bindings
python-gobject-dbg - Python 2.x debugging modules for GObject - transitional package
python-gobject-dev - Python 2.x development headers for GObject - transitional package
python-pyudev - Python bindings for libudev
python3-gi - Python 3 bindings for gobject-introspection libraries
python3-gi-cairo - Python 3 Cairo bindings for the GObject library
python3-gi-dbg - Python 3 bindings for gobject-introspection libraries (debug extension)
python3-pyudev - Python3 bindings for libudev

Since you are looking for python 3 I would install python3-gi, that's probably the package you are looking for, so ...

sudo apt-get install python3-gi