Is there a way to modify the gmail unread badge to show the number of new emails since last open? [migrated]

Currently, when Unread message icon is enabled, gmail will display the total number of unread emails in your inbox.

I personally cannot be bothered to clear out my inbox, so this number becomes completely useless to me.

Is there a way change this badge to display the number of new emails since the tab was last opened, or maybe even just unread emails from Today?