Is thin hardboard an adequate foundation for mosaics? The key factors to consider are where you'll display the mosaic and the size of your tesserae.
Hardboard makes a good foundation for your mosaics as long as you limit the size of the overall mosaic,
limit the tesserae size, and don't display the mosaic in a wet environment. Avoid hardboard for outdoor applications because of the potential for deterioration. Assuming your tesserae are the size of a quarter or less and the overall size of your mosaic is less than 24"x24", I have found that 1/8-inch thick hardboard provides an adequate foundation. If your tesserae are small, it's surprising how flexible the mosaic is, even with grout, which means it can withstand some warping before the grout cracks or glass pieces pop off. If your tesserae are big or if you incorporate large pieces of stained glass into your mosaic, the thickness of your foundation must be greater because the mosaic can't withstand as much warping (i.e., the thicker the wood, the more resistant to warping). For example, suppose your mosaic is 24"x24" and you use a single piece of yellow stained glass to represent the bright sun lighting up the world. Suppose the sun's diameter is 10 inches, which makes up a good chunk of the mosaic. It's easy to see how a little warping can stress that single piece of glass causing failure (i.e., breaking, popping off). It's like ceramic tile on a concrete-slab foundation. As the concrete cracks and moves, stress is applied to the ceramic tile and, if the stress is great enough, the tile breaks. Therefore, you must consider the tesserae size when choosing the thickness of your mosaic's foundation.
Over the years making many wall mosaics that are 24"x24" or less, I have found that my favorite foundation is 1/8-inch hardboard. It's the dark-brown stuff that pegboard is made from but without the holes. It's slippery smooth on one side and rough on the other. I use this material only for dry, indoor, wall mosaics that will not be exposed to moisture. I use this material because it's: 1) Relatively thin, 2) Relatively lightweight, and 3) Rough on one side so the glue grabs hold of it well.
The 1/8-inch thickness allows the finished mosaic to fit in a standard pre-made frame. My glass tesserae are about 1/8-inch thick, so the total thickness of the finished mosaic is only about 1/4-inch. This allows me to buy a ready-made frame for almost nothing. I plan my indoor wall mosaics to be 16"x24", 18"x24", or 24"x24", which are common sizes for pre-made frames. If I were to use 3/4-inch plywood or MDF as the foundation, I would then have to use a custom frame with enough depth to cover the entire thickness of the mosaic (i.e., 3/4-inch wood foundation plus 1/8-inch tesserae equals almost a 1-inch thickness). Custom frames cost up to five times more than standard pre-made frames. For example, by taking advantage of their biweekly 50% sale at my favorite hobby store, I can get a pre-made 18"x24" frame in a lovely style and color that best suits the mosaic, have the mosaic installed in the frame, have the hanging wire installed, and have paper backing installed, all for less than $25. That's right! Less than 25 bucks. A custom-made frame might cost as much as $150.
Not only do I save on framing costs, the hardboard is cheap compared to 3/4-inch plywood and MDF. I buy a pre-cut section of hardboard instead of a full 4'x'8 sheet. The pre-cut section is 24"x48". Knowing the height of my indoor wall mosaics is typically 24" (which is the width of the pre-cut section), this allows me to cut the hardboard giving me a 16", 18", or 24" width for my mosaic foundation. For example, suppose I want my mosaic to be 18"x24". The pre-cut width of the hardboard I buy is 24". I measure and cut 18", which results in a piece of hardboard that's 18"x24". The piece fits perfectly in a standard 18"x24" pre-made frame. I measure and cut the hardboard using a standard circular saw and a "rip fence" that I make by clamping a 3-foot level to the hardboard with two C-clamps. The rip fence allows me to push the saw along the straight edge of the level to ensure a straight and accurate cut.
I prepare the hardboard foundation by painting it with two coats of white primer. The main reason for painting it white is to get a white background onto which the glass tesserae will be adhered (Note: I always adhere the glass to the rough side of the hardboard). Although I usually use opaque glass, the white background helps brighten it up. The dark-brown color of the hardboard makes the glass pieces appear dull and dark, even though the glass is supposed to be opaque. The secondary benefit of painting the hardboard with primer is that it seals it. I don't know if sealing hardboard does anything, but it makes me feel better believing it's sealed. I don't know the material or chemical properties of hardboard and how it's manufactured, so I don't know if it needs to be sealed, but painting it gives me a nice, warm-and-fuzzy feeling. I have a habit of sealing everything whether it needs it or not.
After applying the tesserae and grout, you'll be surprised at how flexible the mosaic is without causing glass or grout failure (assuming your tesserae are relatively small). When I first used 1/8-inch hardboard as the foundation for a mosaic, I experimented and found that I could bend the mosaic a full two inches without affecting the glass and grout. I was too afraid to bend it more than two inches! After the experiment, I assumed if the mosaic can bend a whopping two inches, then it can survive any warping that might occur. Then, after the mosaic was installed in the pre-made frame, I realized that the mosaic was installed in such a manner to inhibit any warping at all. The mosaic was pressed and held in-place with the little fasteners in the back of the frame to keep it from falling out. The only way the mosaic can warp is if it's strong enough to cause the frame to warp with it. I've never had a problem with any indoor wall mosaic warping when using 1/8-inch hardboard installed in a standard pre-made frame.
1/8-inch hardboard is also lightweight enough so the weight of the overall mosaic isn't so heavy that you have to remodel your home to create a support structure stout enough to hold the weight of a mosaic. Generally, my 24"x24" (or less) mosaics are light enough to adequately hang by means of a picture hook and nail installed in drywall. I don't have to cut into the drywall to install 2"x4" pieces between the studs and then replace the drywall. This is extremely advantageous, especially when selling or giving away the mosaic (i.e., you won't lose customers that you might otherwise lose if you tell them they have to hang the mosaic by doing something more than pounding a nail into the wall).