So far as I can tell, apple pie bed is a synonym for short sheet.

According to E. Cobham Brewer's 1898 The Dictionary of Phrase and Fable, an apple pie bed is

A bed in which the sheets are so folded that a person cannot get his legs down; from the apple turnover; or, more probably, a corruption of "a nap-pe-pli bed." (French, nappe pliée, a folded sheet.)
According to wertperch, it's a favorite occupation at English boarding schools and youth hostels, to apple-pie someone's bed!