on an empty stomach

"Drink/eat something on an empty stomach" means "drink it before you eat or drink anything else". You could just about substitute "with" for "of" but it would sound less natural. You can consider "on an empty stomach" to be a set phrase.
 
You could say 'with an empty stomach', but 'on an empty stomach is the much stronger collocation. Look at this Google Books Ngram; the 'with' version is becoming more common, but it's still about ten times less common than 'on'.
 

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