Solution

The new volume is two times the square root of two, in cubic feet.  That is: approximately 2.83 cubic feet.

Explanation

To double the surface area of any regular polyhedron such as a cube, you must increase the lengths of its edges by a factor of the square root of two, because the surface area is the edge length squared.  Hence the volume increases by a factor of "the square root of two, cubed," which is two times the square root of two.

Calculating the new edge length is not necessary because for any regular polyhedron (as well as for any three-dimensional object, so far as I am aware), an increase in surface area by any factor x is always accompanied by an increase in volume by a factor x^(1.5).