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.)