How much do roof racks hurt gas mileage?

ConsumerReports.org has done the analysis. Obviously it'll be different for every car and shape and size of roof rack, and contents of the rack, but in their experiment:

2013 Honda Accord (4-cyl.)       MPG (Miles Per Gallon) @ 65 mph  = 105 km/h
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No rack                             42 mpg     =  5.6 l/100km
Empty rack                          37 mpg     =  6.4 l/100km
Empty rack and wind deflector       35 mpg     =  6.7 l/100km
Rack with two bikes and deflector   27 mpg     =  8.7 l/100km

So that gives you an indication that it can add close to 50% to your fuel bill, depending on the load.

Short answer: It depends tremendously on who you ask and what vehicle is used.

An empty rack has been reported to increase the cost of fuel over time from anywhere from 1% to 15%, while a fully loaded roof rack will add 20% to 50% to your fuel bill. I'll justify these numbers below.

Note that however the exact same sources put the effect of "driving behaviors" at a shocking 35% - so the difference in calming down, using cruise control, and being in a relaxed travel mood while avoiding lane changes is about the same as putting on a roof rack and stuffing it full of bags, depending on your vehicle.

Methodology and citations below.

Sources: Consumer Reports - While this is reported in another answer, it solely uses MPG figures which are terribly bad for cross-comparison, and the empty-roof rack effect seemed so ridiculously high that it made me suspicious. Reports on different vehicles reported at little as a 1% reduction with an empty rack, so YMMV.

Edmunds.com - A multi-vehicle comparison with lots of assorted tests are run, and the data appears to be correctly interpretted into % savings.

Using a 2013 Honda Accord and data provided by Consumer Reports, an empty roof rack will increase the amount you spend on gas by 12% compared to no rack at all, assuming highway speeds. A two-bike rack with wind deflector will increase your gasoline costs by 36%. I am somewhat suspicious of this first figure, but it's possible.

Using a 2008 Buick Enclave with data from Edmunds, empty rails add only 1% to fuel costs but one suitcase and a cooler gave a 21% increase in fuel costs.

I very much like the Edmunds article for it's comparison of driving behavior, A/C usage vs rolling the windows down (verdict for the truck they tested: windows down better than A/C usage), however there still is little comparison amongst vehicles in the same conditions. In my 2010 Chevy Cobalt, for instance, I find A/C has an indiscernible effect on gas mileage possibly due to how the car is engineered, but this is not with scientific testing principles at all.

Final words:

There appears to be a definite effect of loading luggage onto the roof of a car, and it is very likely in the neighborhood of 20-50% regardless of vehicle.

The reason is quite logical - the top of a vehicle is the worst possible place to put things, from an aerodynamic point of view. It tremendously adds to forward-facing surface area that directly hits the wind, it completely screws up the entire wind envelope for the rest of the vehicle, and it does this usually less than 1/3 of the distance into the length of the vehicle; you've basically completely thrown out all aerodynamic gains in the engineering of the entire vehicle, you might as well drive a giant cube.

On the other hand, what if you are already driving a big cube which isn't terribly aerodynamic to start with? Well, I just don't know, but I'm sure putting things on top will make it worse.

However, driving behavior can clearly have an effect as big as a roof rack, so don't stuff luggage inside the cab, annoy the inhabitants, and make yourself want to get there faster - you'll end up losing a lot of money that way!

As to exact or even more precise effects than this rather large range of 20-50%, it appears that bigger, heavier, stronger-engined vehicles are likely to show a smaller % change from roof-racks than smaller, more aerodynamic, lower-powered engine vehicles.

For short trips it probably won't matter, as your gas bill won't be all that high anyway...but if you are talking about a cross-country tour, that little roof rack can cost you a few hundred $ pretty quick.

The answers posted are in line with my experience. I recently traveled in a 2014 Audi A6 3.0 TDI on a trip to pick up my daughter at college. Round trip was about 800 miles.
On the roof was a Rhino Rack with oval shaped cross beams and on the rack was a mountain bike and Yakima box filled with items such as duffle bags, books, etc. Inside the car were more items plus my wife and daughter, and myself and the rear trunk was full.

On the way home over a distance of 400 miles, the vehicle averaged 33.5 mpg on mostly freeway driving at 60-65 mph using cruise control but some smaller roads with a few towns to go through, down from the normal 40 mpg that I have typically measured on long trips.
That is a reduction of about 16% from the 40 mpg, which is close to the range of 20-50% reported by others.

Consider how many pounds of force (which is proportional to horsepower) is required to move the roof top box through the air at cruising speed. Assuming for now that the additional drag is the same for all vehicles (it probably isn't), the box will increase the fuel consumption (in gallons per mile or gallons per hour) at a given speed by the same amount for all vehicles regardless of their size or fuel economy. While the additional amount of fuel is the same, the increase in consumption will appear as a larger absolute change to gas mileage for an economical car than for a gas hog, but it's the same amount of extra fuel and thus the same extra cost (for problems like this it's easier to calculate the European way, consumption per unit distance). In real life I expect a car design that is optimized for low drag will experience a drag increase larger than that of the box itself because the box interferes more with the car's air flow than it would with a vehicle that has a higher drag coefficient, but this gives us a place to start.

Now for some numbers. Last year we made two cross country trips ~ 20,000mi in a 6cyl. 2006 Subaru Outback wagon mostly on interstates at 70-80mph, AC on. We carried a somewhat streamlined Thule box of ~ 3 ft width for on the first trip and no box on the second. We kept careful mileage records (total consumption/miles for each trip), calculating ~ 23mpg with the box and ~26mpg without it. Convert this to the difference in gallons per mile and you can apply it to to obtain a rough extimate for how a box would affect your car. Assuming that a gas engine consumes ~ .45 pounds of fuel per horsepower-hour one can calculate the power needed to push the box. (Diesels might realize ~ 0.40 lb/hp-hr.)

In comparison, we once drove cross country with a 3ft. wide 16ft. canoe atop a minivan - mileage was unchanged indicating that the canoe's improvement of the vehicle's drag coefficient just offset the additional frontal area of the canoe.

Two road bikes on my car roof increased the fuel used by about 50% on a country trip I frequently go on. Normally, I get around 1200km (up to 1350km) range with my 2009 Ford Mondeo 2.0 diesel hatchback. However, I only traveled 810km before having to refuel. Very disappointing but still worth it. Fuel cost was about $90 instead of $60.

I have a diesel car with 2L engine, when I drive with 4 bikes on the roof my fuel increases from 6L/100Km to 8.5L/100Km at around 110-120 km/h cruising speed.

I have find out is better to avoid highways so I can go slower in order to decrease the fuel consumption and temptations to accelerate and drive at a higher speed :)

For example driving with 100 km/h will decrease my fuel consumption with 4 bikes on the roof to 7L/100Km

I've used a Packasport 60 on my 2012 Acura MDX on a stock roof rack since I purchased it, so 8 years of use now. The Packasport looks pretty aerodynamic, and compared with other's results, perhaps it is. Based on my measurements the drag reduces my U.S. MPG by 20% when empty when I am doing about 50% freeway travel. If I am only traveling at slower speeds, around town, it reduces my MPG by about 12-15%. The Packasport has seen a lot of use and is now oxidized and cracked on top, so not as smooth a surface as originally purchased. This hasn't translated into a significant difference in MPG reduction. Without the Packasport I get about 16 MPG around town and 22 MPG on the freeway. The first four years I left the Packasport on all the time, but now take it off when not in use.

I also have a hitch carrier that folds up. When folded up it reduces my gas mileage by ~3%, which I mostly attribute to its 60 lb weight. When folded down it reduces my gas mileage by ~4%.

In the coming year I plan to upgrade to a Lexus 450 (hybrid) for the better gas mileage and will probably only use the hitch carrier.