The Horse can be obtained for free from the Animal Shop. You will need 1,000 reproductions to be able to get it. The Horse reduces your combat cooldown by 20% after completion of a fight.
There are 2 more variants of the horse: the Light Horse and the Dark Horse.
In order to get the Light Horse, you will need to feed your regular horse 1,000 light mushrooms whilst having achieved 3,000 reproductions. Your hero cooldown will be decreased from the regular 20% to 30%.
The reproduction requirement for the Dark Horse is equal at 3,000 but it needs 1,000 dark mushrooms instead. 55% of your regular hero cooldown period will be chopped off, but there is a 50% risk that the Dark Horse will ignore you after a fight. In that case, your usual cooldown time will be in place.