Answer is NO, it does not double.
Step by step.
Equation for surface area (SA) = 2lw+2lh+2wh
If we double the length and width, making them each 2l and 2w, we then have:
2(2l x 2w) + 2(2l x h) + 2(2w x h)
= 2(4lw) + 4lh + 4wh
SA = 8lw + 4lh + 4wh
If the surface area had been doubled, we should get the equation (4lw + 4lh + 4wh) for a doubled surface area.
So no, it is not doubled.