Buried Zener voltage references like LTZ1000A are known for their low-noise. High stability of reference voltage is often improved by temperature stabilization at somewhat elevated temperature.
I wonder, does it impact noise performance? Is there any benefit in lowering operating temperature to room temperature/0/-20/-40°C with the goal of reduced noise, assuming that absolute precision of output voltage is less important than noise?
It might actually be the other way around: The hotter an avalanche breakdown reference diode is, the lower its noise. This paper by Vishay explains it quite well. To summarize: The breakdown of an avalanche diode is stochastic and not continuous. The diode's capacitance gets charged over time, causing the voltage across the diode to increase, until the next electron avalanche carries away some of the accumulated charge. At higher temperatures, those avalanches occur more frequently. Therefore, if the current through the diode stays constant, the amount of charge carried away by each avalanche gets smaller, which also makes the voltage fluctuations smaller.
Of course, if you need low noise, you can just whack a large ceramic or film capacitor in parallel to the reference diode and you're good. Capacitors are as low noise as it gets.
Additionally, as explained in the answer by tobalt, some references might actually get noisier as temperature rises, especially those that contain active circuitry like the LM399. The LTZ1000A is a raw zener diode, however, and should therefore become less noisy with increasing current and/or temperature. (The datasheet does confirm that it becomes less noisy as current increases, so avalanche noise effects should dominate in this case.)
i think Jonathan has it right and what I wrote below applies only to the LM399
There is a plot that shows exactly that in the datasheet of the LM399. When it is cooler, it is less noisy.
However, maintaining temperature stability requires it to be above ambient (when using a simple indirect heater) and stability is vital for low deep 1/f noise.
If you're working in -40C ambient and keep the reference at -30C noise will be lower. Another possible way would be to use a peltier element to stabilize the Zener at a temperature independent of ambient temperature..but none of these approaches are usual or easy and I know no commercial product that does this.
As shown in the other answer by Jonathan S., The Zener breakdown can become inconsistent at low currents and probably more so at very low temperature. Therefore I assume that you need more and more current to tap the low noise potential of lower temperatures.
