A metal wire of Youngs modulus Y is stretched by a load such that the strain is x. If the load is increased so the strain doubles (still within elastic limit), how does the elastic energy stored per unit volume change?

A typical stress-strain curve: the initial straight line (Hooke's Law region, slope = Young's modulus) ends at the elastic limit; beyond the yield point, deformation becomes permanent, peaking at the ultimate tensile strength before the wire finally fractures.