Agree with Dave, cardboard will rot in no time providing zero protection and will be a mildew factory.
Your pool is about 154 sq ft, so if you wanted sand at 1" that's about 13 Cu ft of sand (probably need at least 1.5 times that). You could see how much that would be to get delivered but probably not the cheapest.
Determine the amount of Styrofoam needed and price it locally. Comes in 4x8 sheets, but there is waste and need good tape for the seams.
There is also Gorilla pad which I'm guessing will be the cheapest for your size pool at around $60 on Amazon. Probably wont be the softest, but should protect against what you are describing.