A lot of people think that anticoagulants work by actually thinning the blood. This isn't so.
Rather, these drugs cause the blood to be less "sticky," making it harder for clots to form. The blood remains the same thickness and the amount of blood in the body remains the same.
I think the "feeling cold" happens for two reasons.
One is that it's part of the way your body reacts to the trauma of surgery.
The other reason is the icing - your body perceives cold coming from an external source and reacts by telling you that you're feeling cold. That's your body reacting to an external stimulation.