Google's page says that their toolbar works in Firefox version 2.0 - 4.0.*.

Old browser versions are not necessarily secure, so you should always upgrade to the newest version for the best security. Look for add-ons -- there are many -- that replace the toolbar functionality, like Googlebar Lite.

It was Google that dropped their Google toollbar extension for Firefox (Firefox 4.0 was last Google supported) as Mozilla had nothing to do with that. Mozilla also does not make Firefox just be compatible with the thousand extensions out there, especially with old buggy ones.

http://googletoolbarhelp.blogspot.com/2011/07/update-on-google-toolbar-for-firefox.html

http://support.google.com/toolbar/bin/answer.py?hl=en&topic=15356&answer=1342452
https://support.mozilla.com/en-US/kb/google-toolbar-not-compatible-firefox

You can force install it with a extension like compatibility reporter, but some feature may not work and it has some bugs that has existed since.