In very simplified terms, heritability is the amount of difference in an observable trait that can be explained by genetic variation, as opposed to environmental and other influences. So you can see right away the problem we can have with discussing genetics and heritability of behaviors or temperaments in horses: the environmental differences in how horses are raised and handled can be so huge and are incredibly important in how behavior is expressed. Consequently, good heritability estimates require a large population of subjects, very similar definitions of the specific behavior or trait we want to measure, and the ability to control in real time or statistically the environmental differences.
Behaviors are a very complex sequence of events, and heritability of any behavior is likely associated with multiple genes, which might also be linked to genes for other physical and behavioral traits. Since domestication, it seems clear that over many years horses were (and still are) largely selected and bred for the anatomical and physiological characteristics best for their particular environments and uses.
Selection criteria for individuals might have included certain behavior qualities, but it is also possible behavioral propensities just tagged along with the desired physical characteristics. So we have the breeds we see today with characteristic body types and generally accepted temperament differences.
Temperament is generally defined as a set of behavioral tendencies that show up early in life and remain relatively stable over time. However, temperament is acted upon by the current environment so getting at it in order to really understand its heritability can be difficult.
Temperament studies have been done using surveys, observer ratings, and specific controlled test situations. These studies, along with what we recognize about the differences between breeds, certainly show evidence of heritability of temperament. It's just so far a weaker measure than what we have for heritability of specific physical skills, for example jumping ability.
Still, we know there's such great variation in temperaments even within a breed. Selecting for certain traits is also at the whim of the breeder's preferences. A horse that is "too hot" for one person is an ideal candidate for training to another. Similarly, just defining temperament characteristics, or even the specific behaviors that make a horse skilled at his job, can be difficult to do objectively.
The performance of stereotypies is likely one case where we would probably dearly love to understand heritability.
There’s some research evidence of breed differences and reports of familial tendencies to crib. However, we also need to consider that, broadly speaking, there are similarities within breeds as to how horses are raised—think Thoroughbred racehorses compared to draft horses. So we can't eliminate the effect of early management styles within a breed on the development of cribbing, and we know the important role management can play. It is also interesting to note that offspring of parents who perform stereotypies might be more likely to do so themselves, but their stereotypy might be different from that of the parents. With all this information, it is quite likely with stereotypies the heritability is a propensity, not the specific behavior, again leaving room for other temperament characteristics and the environment to play a big role in its actual expression by any one individual.
We know there are genetic influences on behavior and temperament. I think what we have when we look at the horse in front of us is the underlying propensity to behave in a certain way. If we appreciate the role that the environment and we as handlers can have layered on the horse's underlying genetic constitution, then we are more likely to be successful in getting the behavior outcomes we are seeking.