I came across this invented word reading David Brook’s Bobos in Paradise. He credits the phrase to a Rabbi Gershon Winkler who presides over the only Jewish congregation in Missoula, Montana. When you are the only rabbi in the entire city, you have to incorporate all of the different ideas under one tent. There is no Orthodox, Conservative, or Reform, only Flexidoxy.
Brooks defines Flexidoxy (a wonderfully rare word that has two x’s) as the hybrid mixture of freedom with the rigor of orthodoxy. He uses this word to reflect the spiritual life of the new American elites, the bobos. They long for a spiritual structure, but they are careful not to make that structure so rigid as to constrain them or their liberal sensibilities. Hense we have a new religious revivalism that is careful not to criticize or condemn the non-believers.