Contrary to popular belief, frost doesn't damage grass; walking on it does. Normally when you walk across a lawn, the individual blades of grass are elastic and bounce back without damage. Walking on frozen grass causes the grass leaves to break. When this happens the leaf cells rupture and seriously damage the leaves, resulting in tell-tale blackened foot prints for a few weeks.
If you can't keep off frosted grass, maybe consider putting in a hard path instead?