Yes laundry detergent can still be used to wash clothes with after it freezes.
Just allow the laundry detergent to fully thaw out before using the laundry detergent.
But once laundry detergent freezes it might not clean the clothing or laundry as good as it did before it freezes since the components and ingredients of the laundry detergent break down but the frozen laundry detergent will still clean your clothes.
I've used frozen laundry detergent before and It seemed to get my clothes just as clean as it would when it wasn't frozen even though I've heard that it may not clean as well when frozen.