It's tough to base an estimate solely off square footage. As an example, I moved 90 mile north from a 1,500 square foot house to a 3,000+ square foot house (both built in the 40s), and my utilities went down. Both houses have modern HVAC systems (95%+ efficiency nat. gas furnaces, modern AC) and good insulation. One big difference is that the old house had a lot of surface area with a basement and 1st floor in the shape of an L while the new place is almost a perfect rectangle with a basement, 1st, and 2nd floors stacked on top of each other. Additionally, the new place benefits from having more tree shade in the summer, and is located in a bit of valley so there's less wind pushing on the house in the winter.
EDIT: One more thing beneficial about the bigger house is that during the summer we tend to let the top floor get hot during the day, and just spend more time on the lower levels. This allows us to use the whole house AC less than we would otherwise. In the evenings, when things aren't as scorching, we hit the top level with a quick blast (~30 minutes) of extra AC from window units to cool things down in the bedrooms up there.