The Trainer horse is meant as an introduction to Photo Finish™ LIVE, which uses a 24h exhaustion timer between races for each horse. 

You may notice that your race has fired off earlier than it was scheduled - currently, Trainer races will fire off once they fill adequately to run.

However, your horse will REMAIN ON COOLDOWN BASED ON THE ORIGINAL RUN TIME.

This means it will be exhausted based on when it would have run the race originally before the race time was accelerated.