Window AC unit in my house
This would be completely and fundamentally unworkable. Air conditioners must be installed so that heat is pumped to the outside of the structure.

The entry room would become excessively hot and humid. Remember, water vapor condensed on the cooling coils of the unit is conducted to the outside of the a/c unit where some is re-vaporized and some drips as liquid. This heat and humidity could damage the walls of the entry room.

The temperature in the entry room would be much higher than the ambient outdoor temperature and this would greatly reduce the efficiency of the refrigeration cycle of the a/c. And there would be heat flow through the wall of the entry room back into cooled space.