I have 2 pieces of single walled firehose. One is stamped Flax and the other is stamped linen. From the reading I have done firehose only used flax as the material. Cotton never made the grade.
Flax and linen are made from the same material, just a different level of processing is used and that can vary by quite a bit.
The term "linen" has been used to describe the weave and also to describe the "finish" of belting made from cotton.
Just my 2¢ :)