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For example, if I were to supercool water would it's refractive index still be 1.33 or would it be 1.31, the same as water-based ice even though it's still in liquid form?

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Yes, refractive index is usually temperature dependent. I'm guessing here that the index of supercooled water would be different than room temperature water, but also different from ice, because the density is different in each case.

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Yes, index of refraction in general depends on temperature. (Note that it also depends on wavelength.) My copy of the CRC Handbook gives the following for liquid water using a He-Ne laser with $\lambda = 632.80$ nm:

$n = 1.31744$ at 100 C, $n = 1.33306$ at 0 C, with steps of 10 C in between.

So that's the size of the effect: a change of just over 1% over the temperature range of liquid water. My CRC also reports indices of refraction for several other solvents, and it appears to be usually the case that index of refraction decreases as temperature increases.

If the index of refraction of solid ice is 1.31, that's not due solely to the decrease in temperature. My CRC does not have data regarding the various crystal states of ice, but it does have a table for many other crystal solids. The index of refraction for all of those crystals depends on the crystal structure. For the anisotropic crystal structures, the index of refraction also depends on the direction of the light relative to the crystal axes.

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