I recently filled out an application for an apartment, but decided to withdraw and chose not to sign the lease. I paid an application fee that I didn't expect to get back, but now the broker is pressing me to pay half a month's rent to account for the time lost while the apartment was off the market. I looked at the application and there's no clause saying anything about this. I've read other questions (on this website and elsewhere) that mention people not getting their deposits back after canceling applications, but since I didn't put down a deposit at all, I'm wondering whether the broker has any grounds for requesting additional compensation. Thanks!

No, they are just trying to get money from you. This is not even a common practice, And since there is nothing in the form just refuse.

As Chuck says I don't think you can be legally held to this. But if there was a significant amount of time between the application and the cancellation there might be some grounds. In any event, tell them no and let them try to sue.