Accepted answer
Nobody has published a cycle count, and at 20 mg/mL the concentration tells you what a wrong guess costs: a 0.1 mL aliquot holds 2 mg and every microlitre is 20 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 20 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 20 mg/mL and the date, and never thaw a container you will refreeze.
This is one of the few handling questions with a genuinely quantitative literature behind it.
Buffer salts crystallise at different points during freezing. Sodium phosphate is the classic example: the dibasic form crystallises first and the pH of the residual liquid falls by several units. That pH excursion is the real damage in many cases.
Thaw slowly at room temperature or in the refrigerator rather than in warm water. Rapid warming creates local thermal and concentration gradients that promote aggregation.
Aliquoting to eliminate repeated cycles is standard laboratory practice for exactly this reason.
Nothing here is medical advice, and research-use compounds are not approved for human use.
Dry powder tolerates cycles far better than solution does.