Just a guess on my end. The manga hasn't gotten to that yet.
1) Getting to be king sooner. The prince is super spoiled and selfish, and considering that Rebecca can use the king to threaten him occasionally, it seems he's still under his father's thumb, which rankles.
2) I'm thinking his standing as the crown prince gets a bit shaky. Everyone's mentioned before, but he is a horrible choice as the next ruler. The king seems to be making up for that fact by giving him Rebecca and Arth to do the actual hard work of ruling while he keeps to the fun bits. But Rebecca is now just straight done with him, which has already reached the King's ears. Then he gets the shocking realization that Arth is also subtly drawing away from the prince. The king may dote on the 1st prince, but I don't think its to the point he's willing to see the country fall to ruin after his death, which may very well happen if the prince doesn't have super competent people making up for his shortcomings. I'm thinking the first prince gets a hint that he may lose the position of crown prince and decides to off the king before the change can be made official.