Why did Duke Frederick begin to treat Rosalind unkindly in As You Like It?

At the end of act 1, scene 2, following the wrestling match won by young Orlando, Duke Frederick's attendant, Le Beau, warns Orlando that Duke Frederick is "humorous" (meaning that he's being very moody and increasingly angry), that Duke Frederick distrusts Orlando because he's the son of one of his enemies, and that Orlando should "leave this place."
Le Beau also confides to Orlando that Duke Frederick is growing increasingly upset and displeased with Rosalind, the cousin and best friend of Duke Frederick's daughter, Celia.

LE BEAU: But I can tell you that of late this dukeHath ta'en displeasure 'gainst his gentle niece,Grounded upon no other argumentBut that the people praise her for her virtuesAnd pity her for her good father's sake;And, on my life, his malice 'gainst the ladyWill suddenly break forth.

In act 1, scene 3, Rosalind confesses her love for Orlando to Celia, but their conversation is cut short by the entrance of Duke Frederick, who orders Rosalind to leave his dukedom immediately.

DUKE FREDERICK. Mistress, despatch you with your safest haste,And get you from our court.
ROSALIND. Me, uncle?
DUKE FREDERICK. You, cousin:Within these ten days if that thou be'st foundSo near our public court as twenty miles,Thou diest for it.

Rosalind protests that she'd done nothing wrong against Duke Frederick and wants to know why she's being sent away.

ROSALIND: I do beseech your grace,Let me the knowledge of my fault bear with me:. . . Never so much as in a thought unbornDid I offend your highness.
DUKE FREDERICK: Thus do all traitors;If their purgation did consist in words,They are as innocent as grace itself:—Let it suffice thee that I trust thee not.
ROSALIND: Yet your mistrust cannot make me a traitor:Tell me whereon the likelihood depends.
DUKE FREDERICK: Thou art thy father's daughter; there's enough.

Celia tries to intercede on Rosalind's behalf, but Duke Frederick tells her that Rosalind has stayed this long only for Celia's sake, otherwise he would have banished her with her father.
Duke Frederick tells Celia that Rosalind is deceitful and reiterates to her what Le Beau told Orlando about Rosalind being loved by the people and being pitied because Duke Frederick usurped her father's dukedom.

DUKE FREDERICK: She is too subtle for thee; and her smoothness,Her very silence, and her patienceSpeak to the people, and they pity her.

Duke Frederick gives no other reasons for banishing Rosalind, and there's not another word spoken about it in the rest of the play.
It might be that as he was leaving the scene after the wrestling match Duke Frederick saw Celia, Rosalind, and Orlando talking together when Rosalind gave Orlando the chain from around her neck, and Duke Frederick wanted to put a stop to any association among his daughter, her cousin, and his hated enemy's son, but there's no indication of this in the play.
Essentially, Duke Frederick began to treat Rosalind unkindly and ultimately banished her for the spiteful and seemingly insubstantial reasons of being his brother's daughter, being loved by the people for her own sake, and being pitied because Duke Frederick usurped her father's dukedom, and for being what he considered a bad influence on his daughter, Celia.