In 1998, Connes and Moscovici defined the cyclic cohomology of Hopf algebras. In 2010, Khalkhali and Pourkia proposed a braided generalization: to any Hopf algebra H in a braided category B, they associate a paracocyclic object in B. In this paper, we explicitly compute the powers of the paracocyclic operator of this paracocyclic object. Also, we introduce twisted modular pairs in involution for H and derive (co)cyclic modules from them. Finally, we relate the paracocyclic object associated with H to that associated with an H-module coalgebra via a categorical version of the Connes-Moscovici trace.