I am not a grammar guru--but I know "any time" is used differently in the U.S. vs. in Britain.
I did a quick google search on this topic and found this:
http://www.datsplat.com/grammar-any-time-versus-anytime-one-word-or-two-words/
According to the website above, based on Fowler's Modern English Usage (Second Edition), "any time" is always written as two words in Britain, but can be written as one or two words in the U.S.
I live in the U.S. so I have always learned that "any time" (two words) and "anytime" (one word) carry different meanings. The link above does a good job explaining the differences.
My summary and interpretation (U.S. Usage):
Use "any time" (two words) if you want to convey "any amount of time" (e.g. I am so busy that I do not have any time to watch TV.)
Use "anytime" (one word) to convey "any specific time is ok". (e.g. You are welcome to come anytime.)