How many times hotter than the sun surface is the surface of a star the same size but gives off twice the amount of energy per second?

The effective temperature of a star is relative to the fourth root of the luminosity and is contrariwise proportional to the square root of the radius.

L = k R² T⁴

If the radius remains continuous, while the luminosity doubles, the temperature must increase by a factor of the fourth root of two.

If L → 2L, then T → 1.189207115 T

So the answer is approximately 1.19 times.