LA Fitness received a permit to build a new 37,000-square-foot building at 1000 N. A.W. Grimes Blvd. in April. An opening date is not available. LA Fitness is a nationwide fitness gym that offers exercise equipment, racquetball and basketball courts, and pools and saunas. Round Rock's existing LA Fitness is located at 16600 N. RM 620, Round Rock.