Costco Pharmacy Hours. The Costco Pharmacy hours are the same as the normal Costco Warehouse hours, see above. The Costco Home Delivery service is open 24/7. The Costco Home Delivery Pharmacy is available 24/7 to refill prescriptions, order new prescriptions and check prescription status. Monday- Friday: 5:00 a.m. to 7:00 p.m. PT.