In the past,the cooling effect was achieved using menthol. My skin is sensitive to menthol, so I avoid such soaps.
Recently, a synthetic cooling agent has been introduced that is being adopted by some of the artisan soapmakers. one of the first soaps to be released was Noble Otter Orbit. I have a tub and it does provide a cooling effect without the irritation that menthol gives me.
Barrister and Mann Terror was just released on July 10 incorporating both peppermint and a synthetic cooling agent. I love the Excelsior soap base, but plan to skip this scent due to the peppermint. Will warns that this soap is "VERY COLD".
Although I have not used any mentholated soaps, some with which I am familiar are: Chiseled Face Cryogen, Fine Snake Bite, PAA Frost Byte, Stirling Margaritas in the Artic, CRSW Limenthol, Oleo Soapworks Polar, Proraso Menthol and Eucalyptus, and a number of others.