Answers:
要sudo在所有 Shell会话之间共享身份验证(不仅限于特定gnome-terminal实例的选项卡),您需要将此行添加到/etc/sudoers文件中(用于visudo这样做):
Defaults !tty_tickets
来自man sudoers:
   sudoers uses time stamp files for credential caching.  Once a user has
   been authenticated, a time stamp is updated and the user may then use
   sudo without a password for a short period of time (15 minutes unless
   overridden by the timeout option.  By default, sudoers uses a tty-based
   time stamp which means that there is a separate time stamp for each of
   a user's login sessions.  The tty_tickets option can be disabled to
   force the use of a single time stamp for all of a user's sessions.
   [...]
   tty_tickets     If set, users must authenticate on a per-tty basis.
                   With this flag enabled, sudo will use a file named for
                   the tty the user is logged in on in the user's time
                   stamp directory.  If disabled, the time stamp of the
                   directory is used instead.  This flag is on by default.