Here is a simple look on Horses ...
So ... on global, you build a pasture to acquire horses ...
Only cities with a Stables have access to horses. ... gives access to basic "horse", costs 10 gold, 1 horse, 5 materials (+5 HP, +5 run speed)
A city with a Stables may build an "Animal Training Compound"
The Animal Training Compound ... gives access to "war horse", costs 20 gold, 1 horse, 5 metal, 5 materials (+10 HP, + 5 run speed, +4 armor)
A city with a Stables can build a "Racetrack"
Race Track .. gives access to "racing horse", costs 10 gold, 1 horse, 5 materials (+6 HP, +8 run speed)
A city with a Stables and Animal Training Compound may build an "Animal Armorer Shop"
gives access to "heavy war horse", costs 30 gold, 1 horse, 10 metal, 5 materials, (+12 HP, +4 run speed, +8 armor)
A city with a Racetrack may build an "Olympic Monument"
gives access to "prized race horse" , costs X, (+8 HP, +10 run speed)