Below is a 2011 DC federal case wherein a plaintiff unsuccessfully tried to challenge his Global Entry membership denial. It provides a good overview of the legal frameworks potentially available for challenge, and their limits. It also is a good indicator of the large amount of discretion given to DHS/CBP in administration of the Global Entry program. You can read it in the embedded file below or view it directly here: Global Entry Unsuccessful District Court Challenge.
You can find more information about the Global Entry program in the comprehensive article we have written.