It's not necessarily as bad as you make it out to be. Let's say that you have a hemispherical dome covering 4.3 square miles, which I think is what the summary is trying to say. That's a diameter of 3766 meters and an interior volume of about 14 billion m^3, which is something like 17.15 billion kg of air. It's around 1000 joules per degree celcius for each kilogram. So, if you start with a very nasty 45 degrees celcius and get it down to a comfortable 20 degrees celcius, that's 428 terajoules. Obviously Air conditioning is not perfectly efficient. We'll assume an EER rating of 13 for the air conditioning, which may actually be a bit low for a huge commercial system. That's about 38%, so it would take 1.121 Petajoules. Let's say we're powering by gasoline. There's around 120 megajoules per gallon of gas, which translates to around 24 megajoules of electricity per gallon at 20% efficiency. So, that's around 46.7 million gallons of gasoline. Gas is around $2 a gallon in Dubai, so that's around $93.5 million. That's not very much compared to the initial construction costs of such a structure.
That's just the initial cooling, of course, there's still the matter of keeping it cool afterwards. With such a large structure, heat transfer from the outside is almost negligible with proper design. It's a huge number compared to a regular home, but it's very small relative the the massive volume. Then there's the heat generated inside. A typical human puts out around 100 watts of heat just by being alive, then there's all the lighting, cooking, and every other use of power. Guessing a kilowatt of heat generated per person wouldn't be too far off. From the numbers I've found, I'm estimating that they're expecting an upper limit of about 4 million people continuously (180 million visitors per year, guessing they will stay for a week, plus some permanent residents), so that's 4 gigawatts of cooling, or 126 petajoules per year. Going by our previous figures, that's around $10.5 billion dollars per year. That seems like a huge sum of money, but that's only $58 per visitor if they have 180 million per year (and it obviously scales down somewhat if they have fewer visitors).
These numbers are all rough, of course, and use naive assumptions about the shape of the dome, energy consumption, design efficiency, source of power etc. Obviously powering by gasoline would be crazy from an ecological standpoint, but there's an abundance of solar power available there, and the gasoline cost is just a stand-in. The numbers I gave are skewed towards the worst-case scenario, and they're still reasonable. There's nothing impossible going on there. There may be plenty that can go wrong with such a project, but making out the air conditioning in to a near-apocalyptic problem is a bit hyperbolic.