Actually, we do, but it's so little known (I only ran across it myself a while ago) that it hardly even registers in Google search. Try looking for its cousins, and you'll find misogyny (hatred or dislike of women) coming back with a scary 1,640,000 results, misandry (hatred or dislike of men) clocking in at 1,100,000 results, and the more equal-opportunity term misanthropy (something I definitely fall prey to on occasion) yielding 542,000 results. But misothery? A paltry 1,780 results is all you'll get.
A shame, really, since the term itself (coined by Jim Mason in 1993) contains the resulting misery for animals right within its letters. I like this term, I really do. And it reminds me of the dangers when we don't have proper names for things so commonplace, so underlying just about everything, and yet so silent. Remember the world-changing effect when Betty Friedan described the problem that has no name? Naming is powerful, necessary, and exciting too.
The word misothery may have an awful ring to it, but I'm glad I found it, and will be adding it to my arsenal of tools to fight animal and human exploitation. Because these two types of exploitation are linked, with the one enabling and the other perhaps begetting, but definitely reinforcing. And who knows, maybe one day, certain acts of animal use and abuse will be seen more widely for what they really are: hate crimes based on species.
p.s. if you agree that this term is a damn fine one, and deserves a wider audience, please help pass it on :)