Of course a doubling function is not idempotent!

I think the confusion arises because side-effectful functions can be considered as having type

    f :: (RealWorld, OtherArgs) -> (RealWorld, OtherOutputs)

and so for a garage door toggle you have something like

    t :: RealWorld -> RealWorld

where the new state is the old one with the door opened/closed as appropriate.

Now the idempotence condition becomes:

    t(t(world)) == t(world)

but clearly

       t(t(doorOpenWorld)
    =  t(doorClosedWorld)
    =  doorOpenWorld
    != t(doorOpenWorld)
    =  doorclosedWorld

so this is where the notion comes from. If you abuse notation and just say a function of no arguments can be idempotent then you'll get confusion like this.