Do GPU performance improve when rendering at native or half resolution, compared to fixed scaling?
GPUs can efficiently scale an image by an arbitrary amount (within limits - display options fall within those limits by design) either using a 3D rendering operation or as the signal is sent from GPU to the display. Both of these paths have fully dedicated hardware for the arbitrary resizing and are not likely to be optimized for doubling or halving. Both support a technique called "bilinear filtering" though advances in display hardware may provide higher quality.

If a game is run at non native resolution on a laptop (TVs and desktop displays have their own scalers, but laptop displays generally rely on the GPU), one of these two methods will be used. Display scaling is essentially "free", so it is preferred and the lower the resolution the game is originally rendered the faster the frame rate (or the lower power consumption) - with no special performance benefit for, say, doubling. If a 3D operation is required (say, due to a limitation or issue in the display scaler), the scale operation is not free, but is going to go up with the source and destination sizes smoothly, with no special casing at doubling/halving.

The game may look far better at certain scaling factors, but that's up to the user to decide.

Edit: The intuition behind the question is correct: if GPUs didn't have dedicated scaling hardware, doubling/halving could implemented more efficiently than arbitrary scaling, either in shader code or via a slightly simpler hardware design - though the 3D rendering operation is a requirement of graphics APIs and display scaling is a requirement from systems vendors.