at short notice

Without much warning or time to prepare. Somewhat less common than "on short notice." Who do you think will be free to watch the dog at such short notice? When you're on call, you need to be available to come in at short notice. My mother's retired—she could probably come over and help us with this at short notice.

at short (or a moment's) notice

with little warning or time for preparation.

at (very) short ˈnotice

(also at a moment’s ˈnotice) with very little warning; without much time to prepare: In this job you have to be able to work weekends at short notice.