Temperature, other than substrate access, is the most powerful process-parameter available to a baker when she interacts with her culture. Drastic and sudden changes in temperature do cause sourdough microflora to transcribe the necessary stress-response genes from parent to offspring as they divide and conquer, which can have consequences downstream. These sorts of sudden shifts in living-conditions are of less importance in a final-dough mix, especially one that has received a bulk-fermentation that accomplishes the main priorities a baker might be looking for in her dough. (Another way of saying this is, any mutant-strains resulting from the stress-conditions, desirable or not, will die in a couple of generations, anyway, once they go into the oven.)
Suddenly placing any sourdough culture that was already in log-phase into a refrigerator has drastic impact on flavour, shifting the dominant volatile profile to an overall darker, boozier tone (most of which evaporates in the baking process, with very little remaining in the final, volatile headspace). So, what one gets flavour-wise, is a big bunch of nothing, especially in the crumb. But there's always a flip-side: Gluthatione-production is ramped up under prevailing cold conditions as a survival mechanism for most lactic-acid bacteria, which, for pizza-makers, increases extensibility but under normal fermentative conditions and without the corresponding levels of lactate and/or acetate.
