custom gitconfig for the session

I want to load a specific .gitconfig only for the current session.I tried

git config -f ~/path/to/my/.gitconfig

but it only answers by the man page:

usage: git config [<options>]Config file location    --global              use global config file    --system              use system config file    --local               use repository config fileetc.

I also tried

GIT_CONFIG=/path/to/my/.gitconfig

but it is not taken into account.

EDIT: In this question, "current session" means "current bash session".