The temple mount.
My reasoning is that the image of the beast (the abomination of desolation) comes to life and people must worship it. A statue image in the likeness of the beast person.
Which to worship it, it must be seen. Which would put it out in the open on the temple mount grounds. And because of the time constraints involved, the sanctuary building will likely be a minimum size in order to get the temple ordinances going as quickly as possible, so not enough room inside, for the statue image. The beast person though will be going in and out of the temple sanctuary building.
The other reason for it being out in the open on the temple mount, where people stroll today (under armed guards) is that the false prophet, the one who requires everyone worship the image will apparently gather or force people to gather to worship that image - with guillotines not far away for those who refuse. So there is going to have to a lot of room, open space, available.
Which Jesus gave the warning for them in Judea, to flee to the mountains, when they see abominations standing in the holy place.
To know what makes the image comes to life, it is necessary to go to
Ezekiel 28:16-19. The judgement and exposure of Satan. End of Satan as a terror. Which he will be having great wrath for that time, times, half time - to the inhabiters of the earth in Revelation 12:12.
Satan will be cast down to earth - which will happen in Revelation 12:7-9.
Satan will be exposed for everyone to see... at the end of the time, times, half time.
God will bring a fire up in Satan's midst. And brings him to ashes. Ezekiel 28:18. Exposing him - to everyone that has known him in some way - who will be shocked when they see him. Ezekiel 28:19.
That some way - is that Satan will be incarnating the great statue image. And is how people will worship Satan (perhaps unknowingly) in Revelation 13:4.
Satan has to be present visible to everyone at Jesus's return of Revelation 20:1-3. Jesus's place of return is on the Mt. of Olives directly across from the temple mount.
The false prophet and the beast person will be on the temple mount as well.
So putting it all together, the false prophet has the statue image made, and placed on the temple mount grounds, out in the open, not in the sanctuary building.
in-animate, until Satan is cast down to earth. Which Satan incarnates the image, lying miracle by the false prophet, making it come to life and speak.
People will be required to bow to the image. Either they do, or don't and be killed for not doing so.
When Jesus returns, the brightness of his coming causes the image to go up in flames, and brought to ashes, exposing Satan for everyone to see. And after the casting of the beast and the false prophet into the lake of fire. And the armies gathered to fight Jesus fall dead, after witnessing those things. Satan then will be bound and cast into the bottomless pit.