Careful with taxis - don't use the red top ones ("Al Million Taxis") as they don't know where *anything* is. Trust me, I live here. Then again, by March, there could be Über service, who knows!
With a standard "Mowasalat" taxi, make sure the meter is *on*, and state that you are getting out if it's not on. And make sure it is set to the "1" tariff, not the "0" tariff - a typical trick used to overcharge unsuspecting people.
From the airport, you will also get hit with a QR25 surcharge.
In 6.5 hours, with no luggage/bags, at that time, you could just get a taxi along the Corniche to the Museum of Islamic Art (an amazing I M PEI building) or Souk Waqif.
Then maybe go to the Four Seasons in West Bay, have a refreshment by/on the beach.
Go to dinner at Market by Jean Georges at the W Hotel (a few blocks away, a 10/15 minute walk if you are so inclined), then Arabic/Lebanese coffee and cake at Al Mourjan on the waterfront on the Corniche.
Now that you have seen something, and eaten and drunk your fill, roll back to the airport (make sure you find a good taxi, again, and tell them airport **departures area**) and sit in the lounge.
Perhaps, for ethical reasons, you might choose not to go to Villagio Mall. After the devastating fire in May that killed many children at a day-care centre, some residents have decided to boycott until justice is served.