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?
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.