That is amazingly tacky! On one side, I can see that yes, you didn't give thirty day's notice, but on the other, you're putting your horse down!!!! It's kind of hard to give notice for that. I am so sorry you have to deal with this kind of rudeness.
Like others have said, beyond tacky!!!
To my way of thinking even though I am not in your situation the only thing that would be due is for the days she resides there should those days extend into the month of August.
You might have a peek at your board agreement and see if a 30 day notice thing exists then they would likely be in the right. BUT if there is no such thing, then they have no leg to stand on. You pay what you owe and be done with it. If there is no clause, then your claim would stand up in small claims if it came to that.