In an unmodded game, Patch 1.2, Only the Spanish and Gauls can construct:
temple_of_horse_shrine (Happiness bonus 1, Experience bonus 1)
temple_of_horse_temple (Happiness bonus 2, Experience bonus 2)
temple_of_horse_large_temple (Happiness bonus 3, Experience bonus 3)
Only romans can build upon that: if the temple_of_horse_large_temple is constructed when a roman takes the city, the the roman (and only a roman) is able to build:
temple_of_horse_awesome_temple (Happiness bonus 4, Experience bonus 4)
temple_of_horse_pantheon (Happiness bonus 5, Experience bonus 5)
If you are a roman, and take over a temple_of_horse_shrine or a temple_of_horse_temple, you are stuck, since you cannot build the temple_of_horse_large_temple (only Spain and Gaul can build it).
@EnemyofJupitor: If the temple is of the same family, a civ can upgrade it. Its not dependent on the name of the temple, but the type (or family). Only the greek cities can upgrade a temple of victory (greek name nike, briton name andastra) to the last 2 levels. Mars/Ares/Hercules/Donar is the temple of violence family, which can be built by the romans_brutii (to level 5)/macedon (to level 5)/pontus (to level 4)/german (to level 3) civs. That is, only Brutii and Macedon can build a temple of violence all the way to level 5 (Pantheon).