A vacuum pump is optimized to produce vacuum, which is way different
than moving a large volume of air, which is what you need to do here.
Assuming you can move a large volume of air from the basement, yes it
will temporarily cool the living space. But once you move the existing
out of the basement, which doesn't take very long, it then takes quite
a while, hours, for the air in the basement to cool down again.
When I was a kid, we didn't have central air. Sometimes in the summer,
we'd take the blower cover off the forced air furnace and run the
do what you are trying to do. It gave a brief but noticeable cooling
effect. Howerver, after about an hour, the cold air was gone and
maybe an hour later
it was time to open the windows.
The other problem with this is that it draws hot humid air into the
basement, where condensation from cold water pipes, etc may become a
if you did it regularly, it might introduce enough moisture that would
support mold too.