Cockroaches breathe through spiracles which are small openings on the skin. It is the spiracles and not the blood that carries oxygen to each cell of the body through a set of tubes called trachea. The brain does not c
ontrol the breathing through spiracles. Also, the cockroach does not have blood pressure and so cutting off the head does no lead to uncontrolled bleeding. So a cool temperature could allow a cockroach to last about a month without need for its head, as long as they are not infected by mould, bacteria or virus, which could kill them.
