re: Heat in house due to old single-pane windows?

I have a rent house that is doing the same as yours. Single pane glasses and tuned up HVAC. I do keep the shades or blinds closed when hit with direct sun.

I need to trim a large tree overhanging half the house and am afraid of the consequences if I do.

The previous tenant put a small window unit in the master bedroom.

Edit: Have you checked to make sure none of the ducting has come loose? How is the insulation in the house?