Hi Susan,
First Welcome to the ECS chat!
Yes, you can do it on any count fabric you like.
Now the information for this pattern says it was stitched on 32-count linen going ‘over 2’ squares.
You could stitch it on 16-count fabric and stitch it ‘over 1’ square and the finished pattern will be the same size as stitching over 2 squares on 32-count linen.
Just a tip to help you: If you find some cross stitch patterns you really like but don’t want to use the higher count linens such as 28-count or 32-count or 36-count,
Just divide that higher count fabric number by 2 and stitch ‘over 1’ square.
EXAMPLE: I want to stitch a pattern I found but it states the pattern was stitched on 28-count linen ‘over 2’.
I divide the number: 28 divided by 2 = 14
So I can stitch this same pattern on 14-count fabric but I will stitch it ‘over 1’ square ( which is half of ‘over 2’) and the finished size on my 14-count ‘over 1’ will be the same finished size as stitching on 28-count ‘over 2’.
You can do this to figure out on the other higher count
Fabrics as well.
Hope this helps! :)

Are you talking about a different count cross stitch fabric? Absolutely. You will just have a project that will be a different size. When you do a pattern, you can do anything you want. Most of the time I don't work on white because I just love all the colors that are available. You can change a color or colors in a pattern. Cross stitch, to me, gives me freedom to change to what I like. Pam