It's cold for the time of year.
Is it "it's cold for the rest of year"?
No. Let's suppose you say "It's cold for the time of year" in July. In this case "the time of year" means July (or, more generally, summer).
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No. Let's suppose you say "It's cold for the time of year" in July. In this case "the time of year" means July (or, more generally, summer). Normally you expect it to be warm in the summer, but at the moment it is cold (or colder than usual, anyway).