It is proper for the imam to stand for the two khutbas and to sit for a short while in between them.
Ibn ‘Umar said: “When the Prophet sallallahu alehi wasallam would deliver the Khutbatul Jumu’ah, he did so standing, and then he would sit, and then he would stand [again, for the second khutbah] as the people do today.” This is related by the group.
Jabir ibn-Samura said: “The Prophet would deliver the khutbah while standing, and then he would sit, and then he would stand and speak again. Whoever says that he gave the khutbah while sitting has lied. Verily, I prayed with him more than two thousand prayers [including the five daily prayers].” This is related by Ahmad, Muslim, and Abu Dawud.
Ibn abi-Shaibah records that Tawus said: “The Prophet gave the khutbah while standing and so did Abu Bakr, ‘Umar, and ‘Uthman. The first one to give khutbah while sitting upon the pulpit was Mu’awiyyah,” Ibn abiShaibah also records from ash-Sha’biy that Mu’awiyyah used to deliver the khutbah while sitting, when he became overweight. Some of the scholars say that it is obligatory to deliver the khutbah while standing and it is also obligatory to sit in between the two khutbahs. They cite the example of the Prophet and his companions who always did so; however, the fact that they consistently performed an act is not sufficient to prove that it is fard (obligatory).