I have a Tommy Marchioro 120 - big wire top plastic bottom house for my hedgie. In summer, there's no cover. But for fall/winter/spring, I fashioned a fleece cover for the wire part - it has big cut-outs for the two side doors, one top door/heat emitters/light area as well as cut-outs on the one side that faces out toward the light. It helps the heat emitters do their job of keeping hedgie girl warm when it's a tad chilly in the rest of the humans' living space. Also, it has penguins all over it.
When she's in her smaller vacation cottage and it's chilly out, I use an airplane-type fleece blanket and tuck it around the sides of that house.
Neither cage is fully covered by the fleece... I want her to be able to breathe in there
When the hogs still lived in my room, I would cover their cages with a light-weight, dark blanket so they wouldn't be disturbed when I was working on my computer. They were covered for an hour, at most. It didn't seem to affect them at all... most times, I'd hear them crunching and wheeling as soon as the blanket went on.
The only time I cover a cage with a blanket is when I have rescues/rehomes living in the living room or bedroom. I cover the front, top and ends at night so the TV doesn't bother them. The back I leave open for ventilation.
In my old apartment I covered 3 sides plus the top in the winter because of the draft (she has a C&C cage). Now, there's no draft, but I do cover the top because she's such a shy little girl and seems to feel safer with a top over the cage.
But, I'm wondering, am I doing a disservice to her by covering the top of the cage? Is this "enabling" her huffiness, or is it the opposite - is it good for her since it makes her feel more safe?