No, it does not.
In general, surface area is the sum of the areas of each face of a prism. In a rectangular prism, we have length, width, and height.
The area of the bottom of the prism and the top of the prism is the same, and can be represented as l*w. This means that together, it is 2lw.
The area of the left and right faces of the prism is the same, and can be represented as l*h. This means that together, it is 2lh.
The area of the front and back faces of the prism is the same, and can be represented as w*h. This means that together, it is 2wh.
This gives us the equation
SA = 2lw+2lh+2wh
If we double the length and width, making them each 2l, we then have:
SA = 2(2l*2w) + 2(2l*h) + 2(2w*h)
= 2(4lw) + 4lh + 4wh
= 8lw + 4lh + 4wh
If the surface area had been doubled, we would expect
4lw + 4lh + 4wh
No, it is not doubled.