on short notice
Without much warning or time to prepare. Who do you think will be free to watch the dog on such short notice? When you're on call, you need to be available to come in on short notice.
short notice, on
Also, at short notice. With little advance warning or time to prepare, as in They told us to be ready to move out on short notice. The noun notice here is used in the sense of "information" or "intelligence." [Late 1700s]