I have several. They're single walled and I cook with them often.
I have one of these Nalgene SS bottles also. With a mouth opening of just 2 inches and a bottle depth of 7 inches, outside of a survival situation, I cannot see how you could "cook with them often" unless you are warming up water or a watery broth soup based dinner. If you scorch and burn any food in these bottles, it is not going to be easy cleaning and scouring the inside of these bottles...
I can think of a lot better containers to cook with.
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Earth and sky, woods and fields, lakes and rivers, the mountain and the sea, are excellent schoolmasters, and teach some of us more than we can ever learn from books.
John Lubbock