The force of gravity on an object is directly proportional to the mass of the object and inversely proportional to the square of its distance from the center of gravity. In the case of a planet, the force of gravity experienced by an object on its surface is given by the equation:

F = (G * M * m) / r^2

Where:
F = Force of gravity
G = Gravitational constant
M = Mass of the planet
m = Mass of the object
r = Radius of the planet

Density is defined as the mass per unit volume. So if the density of a planet is doubled without any change in its radius, it means that its mass is also doubled, while its volume remains the same.

This means that the mass (M) in the equation remains the same, but the density (d) is doubled. Since density is equal to mass divided by volume, this implies that the volume (V) is halved. Therefore, the equation can be rewritten as:

d = M / V
2d = M / (V/2)

Since the mass (M) remains the same, we can substitute it back into the equation:

2d = M / (V/2)
2d = M / (1/2V)
2d = 2M / V

From this equation, we can see that the density (d) is directly proportional to the mass (M) and inversely proportional to the volume (V).

Now, let's consider the force of gravity equation again:

F = (G * M * m) / r^2

If the mass (M) remains the same, but the density (d) is doubled, it means that the volume (V) is halved. Therefore, the radius (r) in the equation remains the same. So, when we double the density of the planet, the mass (M) remains the same, the radius (r) remains the same, and the force of gravity (F) remains the same.

However, we need to consider the acceleration due to gravity (g), which is the force of gravity per unit mass. Since the force of gravity (F) remains the same but the mass (m) has doubled, the acceleration due to gravity (g) will be doubled as well. Therefore, the correct answer is option D: It is doubled.