Here’s the quick and dirty of what I’ve found:
Spring 1930: Junior College Song
It’s UMaine’s “Stein Song” with words by Alice and Bobbie McCullough
Fall 1932: For HJC is written, but I can’t find any links to songs. It’s supposed to be an original tune by Bernice Branum and A. Gordon Jones.
Fall 1946: Cougar Fight Song was composed by the UH “Co-Ed Club” members: (sophomore) Marion Edwin Ford, Jr ('49) and (freshman) Forest Roland Fountain ('49).
1950s UH Music Instructor, Ed Gerlach ('46 SHSU) and his Big Band from Texas orchestra made a popular big band version. The UH SOH band played it at a performance in 2010.
I’ve never heard the arrangement (love for someone to pick up the Gerlach album for archiving purposes!), but I believe it might be this version can be heard in the 1955 Audio Yearbook in this video around 2:48:
1950s John Perry, an accounting student, who contracted polio also composed a version, but I can’t any info of him or the song. I do find an Irene Perry who was studying accounting at the time (1951-1954).
1962 Recording - UH’s Concert Band recorded “Campus Songs of the Southwest” which, I believe, @UHRedCat’s recording is from
1972 - UT records “Longhorn Band Salutes the Southwest Conference”
- They use an arrangement by famous composer, Jerry Bilik, a Michigan alum. I didn’t know Bilik is a Wolverine, but for the years I’ve heard this song: I’ve always thought of parts of Hail to the Victors. I truly enjoy this arrangement of the CFS, except parts at the end.
Nice montage video of the UH Marching Band (1969, 1972, 1973, 1976, 1977)
1977 - Cougar Marching Band recording from “Sounds of the Southwest”
1977 - Cougar Brass records an album
1979 - Cougar Brass records a second album
1982 - Recording from SMU’s Southwest Jazz [least favorite - kill it with fire 
]
1996 - The Spirit of Houston Cougar Marching Band Album released -
(@Sideline’s posting)
Late 90s Recording - I think this was pulled from the UHAA website from back when I was in school
Early '00s Recording (I think) - basically, a cleaner version of the late 90s one (what was used in EA Sports NCAA games from back in the day)
2002 - The Oregonian (newspaper) ranked UH’s fight song #7
Early 2000s recording incorporating the “C-O-U-G-A-R-S” spell out:
Segue: 1982 Also had “Bobby and Arleen Harden” recorded “Country Cougar” album in Nashville of one-off songs.
- I’ve never heard it, but if it’s like the Penn State album: it’s not great.
Quick history on the Band:
Out of all the bands in Texas, UH has always been the most Big10 influenced because of Dr. Bill Moffit coming from Michigan State. I’m glad I got to meet him and have him conduct us at one of the games in Fall 2004.