Jurisdiction, perhaps? One of its definitions is 'the power of administering law or justice'. Also, one sometimes hears an accused - in the international court in The Hague, for example - claim: "I do not recognise the court's jurisdiction."
In both The Oxford English Dictionary and Chambers, one meaning of 'cognizance' is 'jurisdiction'. In legal terms, they are effectively synonymous, in other words. (However, if it was a law lecturer who offered the former, perhaps you'd better accept that!)
in ancient legalese the right of a court to hear evidence was known as a comission of oyer, and the right to dispense sentences was a commission of terminer.