How do I run GUI applications as root by using pkexec?

It can be done by adding custom actions to policykit. If you want to run gedit as root with pkexec you have to create new file /usr/share/polkit-1/actions/org.freedesktop.policykit.gedit.policy for example:

<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE policyconfig PUBLIC
 "-//freedesktop//DTD PolicyKit Policy Configuration 1.0//EN"
 "http://www.freedesktop.org/standards/PolicyKit/1/policyconfig.dtd">
<policyconfig>
    <action id="org.freedesktop.policykit.pkexec.gedit">
    <description>Run gedit program</description>
    <message>Authentication is required to run the gedit</message>
    <icon_name>accessories-text-editor</icon_name>
    <defaults>
        <allow_any>auth_admin</allow_any>
        <allow_inactive>auth_admin</allow_inactive>
        <allow_active>auth_admin</allow_active>
    </defaults>
    <annotate key="org.freedesktop.policykit.exec.path">/usr/bin/gedit</annotate>
    <annotate key="org.freedesktop.policykit.exec.allow_gui">true</annotate>
    </action>
</policyconfig>

Finally pkexec gedit should work as expected.

---

Visit manpage or Reference Manual which explains it with EXAMPLE like:-

$ man pkexec | grep -i ^Example -A 60
EXAMPLE
       To specify what kind of authorization is needed to execute the program /usr/bin/pk-example-frobnicate as
       another user, simply write an action definition file like this

           <?xml version="1.0" encoding="UTF-8"?>
           <!DOCTYPE policyconfig PUBLIC
            "-//freedesktop//DTD PolicyKit Policy Configuration 1.0//EN"
            "http://www.freedesktop.org/standards/PolicyKit/1/policyconfig.dtd">
           <policyconfig>

             <vendor>Examples for the PolicyKit Project</vendor>
             <vendor_url>http://hal.freedesktop.org/docs/PolicyKit/</vendor_url>

             <action id="org.freedesktop.policykit.example.pkexec.run-frobnicate">
               <description>Run the PolicyKit example program Frobnicate</description>
               <description xml:lang="da">Kør PolicyKit eksemplet Frobnicate</description>
               <message>Authentication is required to run the PolicyKit example program Frobnicate (user=$(user), program=$(program), command_line=$(command_line))</message>
               <message xml:lang="da">Autorisering er påkrævet for at afvikle PolicyKit eksemplet Frobnicate (user=$(user), program=$(program), command_line=$(command_line))</message>
               <icon_name>audio-x-generic</icon_name>
               <defaults>
                 <allow_any>no</allow_any>
                 <allow_inactive>no</allow_inactive>
                 <allow_active>auth_self_keep</allow_active>
               </defaults>
               <annotate key="org.freedesktop.policykit.exec.path">/usr/bin/pk-example-frobnicate</annotate>
             </action>

           </policyconfig>

       and drop it in the /usr/share/polkit-1/actions directory under a suitable name (e.g. matching the namespace of
       the action). Note that in addition to specifying the program, the authentication message, description, icon
       and defaults can be specified. Note that occurences of the strings $(user), $(program) and $(command_line) in
       the message will be replaced with respectively the user (of the form "Real Name (username)" or just "username"
       if there is no real name for the username), the binary to execute (a fully-qualified path, e.g.
       "/usr/bin/pk-example-frobnicate") and the command-line, e.g. "pk-example-frobnicate foo bar". For example, for
       the action defined above, the following authentication dialog will be shown:

           [IMAGE][2]

               +----------------------------------------------------------+
               |                     Authenticate                     [X] |
               +----------------------------------------------------------+
               |                                                          |
               |  [Icon]  Authentication is required to run the PolicyKit |
               |          example program Frobnicate                      |
               |                                                          |
               |          An application is attempting to perform an      |
               |          action that requires privileges. Authentication |
               |          is required to perform this action.             |
               |                                                          |
               |          Password: [__________________________________]  |
               |                                                          |
               | [V] Details:                                             |
               |  Command: /usr/bin/pk-example-frobnicate                 |
               |  Run As:  Super User (root)                              |
               |  Action:  org.fd.pk.example.pkexec.run-frobnicate        |
               |  Vendor:  Examples for the PolicyKit Project             |
               |                                                          |
               |                                  [Cancel] [Authenticate] |
               +----------------------------------------------------------+