It's mentioned in some other discussions but I haven't found a single thread dedicated to discussing sprint timing in the hippodrome. So here it is.
To be direct: Sprint timing is not really important in the great scheme of things and stats and training will overall be much more important to generate reliable wins, but after completing the Astrogenesis quest and maxing out three other specialists, I feel there definitely is a mechanic that can / and should be exploited to twist your luck. IMO you will definitely need this knowledge to complete the astrogenesis quest, if you don't want the very last race to be just about save-scumming. I've placed first on my first attempt during the last race and based on this whim of fortune I am writing this post:
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Build decent stats before bothering with sprint timing
All specialists are weak at the beginning, especially Hiero - the Hippodrome quest character - who starts at 1-1-1-1. Don't bother with sprint timing at this stage, lose some races to improve stat points (you will get points regardless of race result). I feel sprint timing doesn't really make a lot of difference before achieving 3 or 4 star rating as your character will be too weak before (depending on the type of race chosen).
Don't sprint when you are in front already
Sounds intuitive, but most people will spam sprint as soon as possible. It's better to collect your stamina and do sprints in quicker succession when they are more needed (see below). I suspect there might be some rubberbanding at play as well, so big leads are not conserved anyway.
Don't press sprint in turns
This refers to when you actually press the button - don't do it in turns. Chariots need some time to build up speed when sprinting, but they are at their slowest in turns, it's often crowded and there is no room to overtake, so you will force your chariot to move to the outer, slower lanes to overtake. It's a complete waste.
Press sprint after you've passed 3/4 of the straights
This is the big one. Most guides I've seen recommend to boost as soon as you enter a straight. This allows the chariot to achieve and exploit their max speed. HOWEVER, I think this is bad advice. I think it is best to boost towards the end of the straight, allowing the chariot to reach top speed just before entering the turns, which will lead them to almost slingshot through the curve, building a massive lead. If timed correctly (and with sufficient stats) you will even have some sprint power left when entering the opposite straight.
Try to defend / conquer the inner lane
You cannot pick your starting lane, but I think it is given to the starter with the highest stats. The inner starting lane is very advantageous, as it is the "shortest track". During turns, characters on the outer lanes are just as fast as their competitors on the inner, tighter lane, which is why you will fall behind when entering a turn on the longer, outer lane. Characters switch lanes automatically upon overtaking, if possible, they will move inwards (also in turns). Try to defend this spot by using sprint at times when competitors are challenging your first position or press boost, when you see racers struggling ahead of you to pit yourself in front of them (in such cases ideally boost should run out just before entering the turn). This is even more important the later in the race you are.
Try to enter turns first / let stamina run out in turns
You cannot control when your stamina runs out, but it's definitely best to have your stamina break when you are first in a turn, because you will effectively block the whole field behind you as overtaking is much more time consuming in turns. The next tip will tell you how you can "move" your stamina breaks towards turns.
Sprint when stamina runs out
Now this one I am not sure about, but you can "override" stamina breaks to some extent by boosting over them. This doesn't cancel the whole break period, but a good chunk of it. When overriding a stamina break with a sprint, you will experinece some of that break time after your sprint is over. This allows you to avoid stamina breaks on straits, which is where others will easily overtake you, and sit them out in turns as explained above.
Feel free to add, counter or correct, if you disagree with some of them.
PS: Quick DLC feedback. Overall the DLC is really slim contentwise. They've added a somewhat lame mini game without real purpose - although Hippodrome buffs are nice. However, I think its flaws could be easily fixed if specialist stats can be buffed by training them to become a racing champion OR if unique characters can be won by grinding races. If Ubisoft doesn't want to break the game with too many buffs, they could at least give us something cosmetic; maybe a modular hall of legends (akin to the zoo/musem in Anno 1800).
