Thunderbird is a mail client. You want to sync with the server. If you are saying that you want your mail available to your phone and your computer, you need to set up IMAP email connections rather than POP3. IMAP keeps the mail on the server and lets you view it on your devices.
https://secure.wikimedia.org/wikiped...ccess_Protocol
https://mail.google.com/support/bin/...y?answer=75725