In BCS theory they break particle number conservation and show the existence of a gap, which would explain why groundstate properties stay relatively the same even for higher temperatures (until beta is of the order of the gap).
However, as long as you don't couple to the electromagnetic field, there is no Higgs mechanism and therefore there are also massless excitations along with the Cooper pairs (these massless excitations are fluctuations in the order parameter https://arxiv.org/abs/cond-mat/0607493)
But without such a gap (so without the anderson mechanism), why are groundstate properties still kinda invariant at nonzero temperature? Or maybe better, can you analytically show that after applying a simple potential difference you will have infinite current even with the goldstone-mode corrections?