Autoload is the way to go here.
You could save the skill cooldown value in a file somewhere in the user:// path (you can't save things to res://), but I/O is some of the slowest stuff you can do on a computer. You don't want to do I/O in the middle of the game if you can avoid it. What's even worse is that just saving a single cooldown value (possibly a single float) is extremely inefficient - you'll need to open and close a file just to write something like four bytes to it, and the operating system overhead for opening and closing files is huge. Finally, if you're running on an SSD or some kind of flash memory and your OS doesn't automatically batch writes, you'll degrade your flash memory, especially if you have many scene switches (that's most likely not a problem and rather theoretical, but it's still something to keep in the back of your mind if you plan on doing lots of writes to disk).
Using Autoload instead will keep the data in memory for you and sidestep all these issues.
Only store the cooldown value to disk when you want to quit the game and restart it having the same cooldown state, and store it with a bunch of other stuff that needs to be saved so the overhead you incur is worth it.
A third option might be to send the cooldown value to a server if you're writing a networked application, but again, this hardly seems worth it just to keep a value when switching scenes. The network overhead is huge, and the network might not be there when you need it. This option mainly applies when you write a multiplayer game which needs to update game state all the time anyway.