Maybe there is an add-on to do it, but I didn't found one.

I know this way (tested on Windows, assuming it is the same for Linux):

1. Install DB Browser
2. Start DB Browser and open places.sqlite file (in Firefox profile folder)
3. Select moz_annos table
4. Click on File > Export > Table(s) as CSV file…
5. Click on OK, enter a file name ans select location

A CSV file is a text file, you can open it with any text editor, or better, with LibreOffice Calc. Each download corresponds to 3 lines: 1) download location 2) file name 3) time and size.

You can also do such queries with the SQLite Manager extension in Firefox.

• https://addons.mozilla.org/firefox/addon/sqlite-manager/

I use this:

SELECT content AS Filename, datetime(dateAdded/1000000,'unixepoch','localtime') AS dateAdded, url AS Location, moz_anno_attributes.name AS Annotation
FROM moz_places, moz_annos, moz_anno_attributes
WHERE (moz_places.id = moz_annos.place_id) AND (moz_annos.anno_attribute_id = moz_anno_attributes.id) AND ((moz_anno_attributes.name LIKE 'downloads/destinationFileURI') OR (moz_anno_attributes.name LIKE 'downloads/destinationFileName'))
ORDER BY dateAdded DESC