Trying to visualize the argument...

First, imagine a star with precisely 2 times the mass of Sun.

Further suppose that it has precisely 2 times the radius of Sun.

And precisely the same radial distribution of density.

This means that at any point its density is precisely 1/4 that of Sun at 1/2 the distance from centre.

Then the gravitational acceleration should be 1/2 that of Sun at surface and any corresponding depth.

The weight of a column of given length would be 1/8, and since the radius is 2 times bigger, the central pressure should be exactly 1/4 that of Sun.

1/4 pressure and 1/4 density might mean the same temperature at centre, and any corresponding depth.

Not quite.

For one, the radiation pressures will be independent of matter density, and equal at equal temperature.

Let us assume that this is negligible for a star like Sun, or even 4 times less dense.

For another, the mass density may be exactly 1/4. But the number density, and therefore ideal gas pressure for equal temperature, would not be.

Because ionization state depends not only on temperature, but density. All atoms are ionized in the centre of Sun, and light atoms like H and He are completely ionized; but heavy ions hold on to inner electrons, and there are electrons which are held by ions in centre of Sun which would be ionized at exact same temperature but 1/4 the mass density. Therefore 1/4 the mass density will mean more than 1/4 particle density and more than 1/4 particle pressure.

Let us assume that this also is a negligible effect, though.

Then is it correct to state that a star with 2 times the mass and 2 times the radius of Sun should have closely equal interior temperature?