No, it does not cost more. The United States Postal Service (USPS) classifies Puerto Rico as a domestic destination. Therefore, mailing a letter or package from any U.S. state to Puerto Rico costs exactly the same as mailing to another state.
UPS and FedEx also treat Puerto Rico as domestic for ground and air services, though they may apply extended area surcharges for remote ZIP codes—just as they do for rural Alaska or Hawaii. USPS does not.
So whether you're sending a postcard from Maine or a care package from California, the postage to Puerto Rico is purely domestic.