Accepted answer
Nobody has published a cycle count, and at 2.5 mg/mL the concentration tells you what a wrong guess costs: a 0.1 mL aliquot holds 0.25 mg and every microlitre is 2.5 µg. Damage from freezing is not gradual attrition — it is concentrated at the phase transitions, where ice excludes solute and the unfrozen fraction climbs well above 2.5 mg/mL for as long as the transition lasts. Two slow cycles can therefore do more than four fast ones, which is why a cycle count is the wrong unit in the first place. The way to make the number one is to make it one: split at reconstitution into single-draw aliquots, label each with 2.5 mg/mL and the date, and never thaw a container you will refreeze.
Answer first: each freeze-thaw cycle costs something through aggregation and pH shift, so the mitigation is aliquoting rather than choosing a better freezer.
Never refreeze a thawed aliquot. The whole point of aliquoting is that the aliquot is single-use, and refreezing it discards the benefit.
The part that matters: the mitigation is aliquoting. Divide the reconstituted solution into single-use volumes before the first freeze, and each aliquot then experiences exactly one cycle.
Aliquoting to eliminate repeated cycles is standard laboratory practice for exactly this reason.
The caveat is that aggregation is invisible in a clear solution below the threshold where it becomes visible.
Thaw slowly and never refreeze an aliquot.
edited 6 Sept 2024 by forty_two_c — reworded for clarity after a comment