UH started SWC play in 1976, and the conference broke up in 1996, and during those 20 years they won 7 times, so no they weren't a true rival. But they did win the SWC in their first year. Their best success was using the run and shoot, which was new, novel, and people didn't know how to defend it.
Also about that time, the had a big cheating scandal. They hired a law firm to investigate what was really going on prove they were honest. They fired that firm announcing nothing had been found. A few weeks later, the NCAA put them on probation. They blamed Texas for turning them in (as opposed to admitting they cheated).
After the SWC broke up, their program had terrible facilities and support. They wanted to bring Football back to campus and want to play Texas in their old tinker toy stadium. Dodd$ wanted to play it at Reliant Stadium, but they insisted and built some bleachers to add capacity. Couple days before the game, and inspector condemned those bleachers . They refused to refund those tickets, rather allowed those ticket holders to watch a closed circuit broadcast.
Now they have hired a rape accessory for their football staff, and want to swing their dicks like they are big boys. Well....fuck them.