To download your Facebook LIVE video, or download a Facebook post that you've shared, you can do this from Facebook's Creator Studio.
Step 1: Navigate to your Facebook page
After you get to your page on Facebook, look on the left sidebar for "Creator Studio."
For this to work properly, make sure that you're an admin or manager of the page you're trying to download a video from.
Step 2: Navigate to your published posts and sort by LIVE videos
Once here, you can view all of your LIVE video posts and manage a lot of options at once.
Step 3: Select the video that you wish to download
After selecting the post, several options will pull up on your toolbar. Select "Download HD" to begin the download.