Correct Answer  Option 2 : 4 times
CONCEPT:

Moment of Inertia: A quantity expressing a body's tendency to resist angular acceleration, that is the sum of the products of the mass of each particle in the body with the square of its distance from the axis of rotation, is called the moment of Inertia.

Rotational energy or angular kinetic energy: The kinetic energy in a body due to the rotation of it. Mathematically expressed by:
\(K=\frac{1}{2}× I × ω^2 \)
where K is the rotational energy, I is the moment of Inertia and ω is angular velocity.
EXPLANATION:
Given that ω' = 2ω
The rotational kinetic energy of the body:
\(K=\frac{1}{2}× I × ω^2 \)
\(K'=\frac{1}{2}× I × ω'^2 \)
\({K' \over K} =(\frac{\omega '}{ω})^2 =(\frac{2\omega}{ω})^2 = 4\)
K' = 4K
 So if the angular speed is doubled the kinetic energy will become 4 times.
 Hence the correct answer is option 2.