I use heavy weight watercolor paper, usually 140 pounds or more, either hot or cold press for my paper pyrography projects. Because the paper can buckle during the burn, I prefer a watercolor block instead of individual sheets. A block is a stack of paper that has been glued along all four of the outer edges, which keeps each sheet in the stack from moving or buckling during work. When your burning is complete you run a sharp bench knife or craft knife under the edge of the top sheet, which cuts that sheet free from the block.
If you do not have a block of watercolor paper you can lightly mist the paper with clean water until it has a soft, damp sheen. Let the paper rest for about 5 to 10 minutes, so it has a chance to expand. Then thumbtack or tape the sheet to your drawing board. Allow the sheet to completely dry before you begin your burning.
You might also consider Stratmores Stonehenge paper series. This line has a fine tooth and comes in several neutral color tones. Pastel papers are great for your work as they also come in a variety of colors. You can even purchase greeting cards and envelopes made with watercolor paper as gifts for your family and friends.
I find that I need a fairly hot setting on my burning unit for paper, I use an Optima. The tonal values are controlled in three ways. First by the temperature setting. Second by the width of the burning pen tip – wider tips fill more area than thin tips. And third by how tightly I pack the detail lines into an area. I use a slow movement for my pen, again I find paper is slow to take a good, strong tonal value burn.