Surely GSX can start with a hotkey, which you can set to any key with the Config panel in the FSDT installer.
However, like each and every other 3rd party Toolbar addon that might need to be closed or opened on the fly (Navigraph is the same), it won't respond to the hotkey if the Toolbar icon is Inactive so, while you can surely open/close it with a Hotkey, if toolbar menu was Inactive the last time you returned to the main menu, when you enter again, you need to first open it with the mouse, THEN you can start opening/closing it with the hotkey.
The reasons for this are fully explained at Page 84 and following of the GSX Manual, the chapter named
The GSX Toolbar Menu, no use repeating the manual entirely here, I'll only quote the final part to suggest how to use it correctly:
This is now your normal workflow should be:
• Open the GSX menu from the Toolbar icon when first starting Ground services.
• Don’t close it from the Toolbar while Ground services are performed, always use the Hot-key or the X icon to close the menu.
• Close the menu using the Toolbar icon when you are done with Ground services, usually before take-off.
In brief, the Toolbar Icon should always be left Active while Ground services are required.