From where does gas oven get combustion air?
Home gas ovens draw oxygen from room air, and in turn add CO2, some CO, a little SO2 (from impurities and warning smell additive) and perhaps some nitrogen oxides to the room air. An externally vented exhaust hood helps get rid of these combustion products.

Most homes are not so air-tight that use of a stove or oven (other than on self-clean cycle) poses an issue. However, there may be local recommendations or requirements for air exchange, but "in the United States, venting most residential gas ranges and cooktops to the outside is not required."

So, if your evaporative cooler is vented, and the airflow interferes with the oven, you may need to operate the range-hood fan or crack a window (admittedly reducing the efficiency of the swamp cooler).