Make your cat a companion it can sleep in, a Catpanion Cube! This project can be done over a weekend, so get sewing!
(Warning: several cute cat photos follow)

Supplies

1 inch thick foam padding (you'll need enough for six equal sized squares)
medium gray cotton or similar fabric. (how much will depend on how big your foam squares are. You will need enough for twelve squares an inch bigger than your foam squares.)
light gray cotton or similar fabric. (This will make the circles on the five sides of the cube)
pink cotton or similar fabric (This will make the hearts, you can also use it on the opposite side of the door hole if you want)
pink ribbon
gray and pink thread
sewing pins
pattern making material or poster board
sewing machine

Step 1: Cutting Out Your Squares

Hopefully you have a nice workspace to cut out your material, or else you might have to wrangle cats off your fabric.

You first need to cut out six squares of foam padding. I repurposed 14 x 14 squares that made a nice sized cube.

Once you have your six foam squares, you can start cutting out fabric squares out of the medium gray fabric. You will need twelve squares an inch bigger than your padding squares, so I cut out twelve 15 x 15 squares.

For the heart and circles on the sides of the cube, make patterns out of poster board or pattern making material and then cut out 5 of each (or four if you don't care about putting one on the bottom of the cube).

Step 2: Doorway

Use the same circle pattern to cut a hole in one of the foam squares that will be the doorway.

Use the foam square to mark and cut out circles in two of the fabric squares.

Pin the two fabric squares together and sew around the diameter of the circle.

Turn right side out and fit around the door hole. Pin the edges around the fabric square.

Step 3: Decorating the Cube

Begin pinning lengths of the pink ribbon across the fabric squares. You only need to do four if you don't care about the bottom side. Sew the ribbon down with a straight stitch.

Next, pin the circles and hearts over the ribbon. Change your sewing machine stitch to the applique or zig zag stitch and sew down. I did the hearts first so I didn't have to change thread after sewing the ribbon down.

Once you are through with applique on all the squares, pin your decorated fabric square around your foam square with one of the blank fabric squares on the opposite side.

Step 4: Compiling the Cube

To assemble the cube, pin and sew the edges together, with the decorated sides facing inwards to form a cube. You want to make the seams are as close the the foam as possible. It can take some finagling to get it situated right under your sewing machine. Make sure the door and bottom are positioned in the right spots.

Once all the sides are sewn together, carefully pull the cube right side out through the door hole.

Step 5: Testing Time!

Hopefully your cats didn't sleep on your fabric and wrinkle it up. I doubt cats will care either way, but you can steam the wrinkles out if you desire.

Don't be surprised if all your cats want to hang out with their new companion.