There are -as far as I know- no "proper" semiconductors (in this case something with a direct bandgap since the question concerns a LED) that become superconducting at low temperatures (you can play around with proximity effect etc, but that is not the same thing).
That said, I don't think there is any known fundamental reason why it would be impossible for a HTS material to be a semiconductor at room temperature and superconducting at low temperatures.
However, it could obviously never be both at the same time since there couldn't -by defintion- be a bandgap in a superconducting material. So I am reasonably sure the answer to your question is no.
At low temperatures, forward biased junction diodes make very linear thermometers. This is consistent with the ideal diode equation.
V = (kT/e) Ln (I/I0)
I = forward current, k = Boltzmann's constsnt, T = temperature in kelvin, and e = electron charge.
Here is a recent paper:
GaAs Junction diodes as cryogenic thermometers
Cohen, B.G.; Goordman, R.V.; Snow, W.B.; Tretola, A.R.
Electron Devices Meeting, 1962 International
Volume 8, Issue , 1962 Page(s): 74 - 76
Digital Object Identifier
Summary: It has been observed that the forward voltage drop, at constant forward current, of a GaAs diffused p-n junction varies almost linearly with temperature from 2.0° K to above 300° K. Since carrier "freeze out", at low temperatures is not observed, these junctions make excellent cryogenic thermometers. In addition these diodes exhibit good repeatability on temperature cycling and an insensitivity to magnetic fields. The sensitivity of the devices measured near room temperature is Δ/VΔT∼ -3.5 mv/°C; If= 0.1 µA Δ/VΔT = - 2.0 mv/°C; If= 1.0 µA The sensitivity decreases slowly toward lower temperatures and at liquid Helium is: ΔVΔT∼ - 1.5 mv/°C and is essentially independent of current. This high sensitivity over such a wide temperature range is unique among cryogenic temperature indicators.
We used to use Si diodes in a tiny SOT23 surface mount package, they had a pair in series so you got twice the output, feed it a 100uA from the micropower circuit example in Horrowitz+Hill. They are tiny and cheap, although you can buy the same thing calibrated(?) from Oxford Instruments for a fortune.
I only ever used them at LN2 - there might be too much power dissipation if you ran them continually at very low temps.
I've used standard SMD GaAs diodes down to 4K, they work but are unpredictable (probably due to the packaging) and they need to be cycled a few times before they reach any kind of stability. Some schottky diodes also work reasonably well.
In fact, MOST components can be used as thermometers. For many years the "standard thermometer" in low temperature physics was a particular type of Allen-Bradley carbon resistor (sliced up into smaller pieces) which just happened to work well. You can still buy them from companies that were smart enough to buy a lot of them before Allen-Bradley stopped making them (but now they cost a fortune).
The main problem is that you still need to calibrate your DIY thermometer somehow.
The advantage of DT470 (which is a Si diode) and other similar modern commercial sensors is that they follow a standard curve (they are made in such a way that they are almost identical), meaning even an uncalibrated sensor is good enough for many applications (something like +-0.2K below 10K). And they are only something like £100-200 which is not bad considering how much time (and expensive) it can be to calibrate a sensor yourself (although a calibrated sensor is significantly more expensive).
At lower temperatures (below say 2K) resistive RuO2 is the most common type of sensor, although e.g. capacitive sensors are also used.
Low temperature thermometry is actually quite tricky, especially when you go down to mK temperatures where self-heating becomes a serious issue.
There was also a 2 wire current source (LM504?) that gives 0.1mA/Kelvin (so it gives 27.3mA at 0C) so it was trivial to hook up as a temperature regulator. Officially it only worked down to -40C but about half of a batch would work down to LN2.
Then the maker tightened up their manufacturing process and none of them worked cold.