Roy Cohn is shown in the play to be the personification of mortal power.  Cohn's fascination with the telephone is linked to the idea that he can "get a hold of" any person around the country, at any given moment with the telephone.  His ability to have people on a "speed dial"- like capacity is symbolic of his power.  It is also symbolic of his need to wield power and use it as a force of intimidation.  The idea that Cohn can "call anyone" is something that he has used in order to achieve what he wants.  It was the idea of using the phone to divulge falsehoods about the Rosenbergs that enabled his career to become viable.  Roy is a creature of power, believing that it is the base of everything.  His embrace of Republican values and governmental authority is merely a front for the power that he covets.  The telephone is an instrument, the instrument for the time, where power can be immediately flexed and used.  Had Roy Cohn been a modern invention, he would have had a Blackberry or an iPad, or some other type of tablet that would enable him to accomplish his power- based desires.  His link to the telephone in the play is one of power, and the need to constantly use it as representative of something as primal  "intestinal churning" or "gastric juices."