The Apache was a joint Rock Island-Southern Pacific train that ran over the Golden State route between Chicago and Los Angeles from 1926 to 1938. A 1937 timetable posted by the Rock Island Technical Society actually shows three trains on that route: the all-Pullman Golden State Limited, the Californian, and the Apache. The Californian operated about an hour earlier than the Golden State Limited and must have been for coach and other budget passengers. The Apache operated about 11 hours before the Golden State.
A 1948 timetable, also posted by the Rock Island Technical Society, shows a train named the Imperial leaving Chicago about 11 hours before the extra-fare Golden State, while a train named the Southwest Express leaves about four hours before the Golden State. So, for some reason, the name Imperial has replaced Apache. For what it’s worth, the Southwest Express has the same train numbers as the Californian, but the Imperial‘s numbers are not the same as the Apache.
According to this ad, as of 1936 both the Golden State Limited and the Apache were “air-conditioned throughout.”