All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
Filming or recording the Friday khutbah [sermon] with a phone is not allowed if it requires moving your hands during the khutbah, because that distracts you from listening to the khutbah. The Prophet, sallallaahu 'alayhi wa sallam, said: “Whoever touches pebbles [during the Friday khutbah], he has engaged in laghw [useless activity or idle behavior].” [Agreed upon by Al-Bukhari and Muslim]
Al-Nawawi said: “This includes the prohibition of touching pebbles or engaging in any form of distraction during the khutbah, and it points to the importance of focusing with both the heart and body towards the khutbah. What is meant by ‘laghw’ here, is what is false, blameworthy, and rejected.” [End quote].
Lastly, if recording it [the Khutbah] does not require moving the hand during the Khutbah, for example, starting the camera before the Khutbah begins and then leaving it alone without being distracted by it, then there is nothing wrong with it.
Allah knows best.