Appealing to the stones to cry out when the disciples fall silent is Jesus' way of affirming that God is on his side. Surely he has a strong inkling that many of those who say "Hosanna" today will be heading for the hills before long. There is a saying variously attributed to Franklin Delano Roosevelt and Nelson Mandela that "courage is not the absence of fear but the willingness to move into a challenge, fear and all." Then there is another saying: "When the going gets tough, the fickle get going." Ok, I made that saying up, but it fits this occasion. There is courage, embodied by the steady, sure course of Jesus' mission and there is cowardice, exemplified by the desertion of the disciples we can expect over the days of Holy Week. People are only fickle when they don't get what they want. If Jesus would be a real king, if he would promise military victory, good fortune, good health and good luck to all his backers, everyone would keep filling the skies with praises all day and all night, all week and all year.
Luke 19:37 tells us that the crowds were "praising God for all the deeds of power that they had seen." They were, I assume, talking about healings and feedings and exorcisms. They hadn't signed on for the slash of a whip or the pounding of a nail. Deeds of power didn't equate in their minds with a spirit of servanthood and humility that would lead to a cross.
It is much easier to pop in for Palm Sunday, and then pop out for Maundy Thursday and Good Friday before popping back in for the trumpets, the lilies, the new clothes and the candy on Easter Sunday.
Sources Cited
William Lane Craig, "The Triumphal Entry"
Howard Marshall, The New International Greek Testament Commentary on Luke