Using sharp knife, cut gluten free puff pastry in half. Re-wrap other half and store in fridge or freezer. Allow to come to cool room temperature, wrapped in plastic wrap.
In small bowl, mix cheese and cayenne pepper. Sprinkle ½ of cheese mixture onto work surface. Roll puff pastry over cheese mixture into a rectangle about 1/8-inch thick. Sprinkle other half of cheese mixture over top half of rectangle and fold bottom half over cheese. Using rolling pin, press and roll pastry out again to 1/8-inch thick.
Cut pastry into 1-inch thick strips using a pizza cutter or very sharp knife.
Twist pastry strips onto parchment-lined baking sheet. Place baking sheet in refrigerator to chill for at least 10 minutes, or up to about an hour.
Preheat oven to 375°. Bake cheese straws for about 17-25 minutes, or until puffed and golden brown.
Serve warm or at room temperature.