The 56 - 57 degree cave temp is about what is normal for middle TN. It does sound like what is known as a cold air trap. This phenomena occurs in other caves; one that comes to mind is Fern Cave in AL, where a cold air trap creates an optimal environment for the huge colony of bats that roost there. I am not personally familiar with what sort of dynamics cause this condition. However, if your newly opened cave has this condition, I wonder if it's an indication of impressive size and/or length, perhaps on the same order as Fern Cave?
You didn't mention how long this cave is, or if there seems to be potential for finding more passage. If at all possible, follow that air! Best wishes!