The new period is 8 years.

Explanation

Kepler’s Third Law (M₁ + M₂) P² = a³ Mass (M₁) of the Sun stays the same, M₂ mass of the Earth doubles, the average distance or semi-major axis is 4x, and we are looking for P the period of the orbit

Since the mass of the Earth even doubled is not near the Sun’s mass, the sum (M₁ + M₂) stays the same.

Inserting 1 for things that don’t change and the change factor for things that do change

(M₁ + M₂) P² = a³ = 1 x P² = 4³ = 4 x 4 x 4 = 64

P = 8 since 8 x 8 =64

Units for this equation are solar masses for M, years for P and AU for a