there is a couple of differnt sources that could be letting in cold air. If this is a direct vent fireplace that is mostly sealed off from access and "breaths" through tubes or pipes that go through your outside wall, then it is likely bad or no insulation around the gas lines or venting pipes. This can be remedied easily once you find the pipes.
If this fireplace has a regular stack chimney and the hearth is accessable to you, it could be that the fireplace damper is clipped open or disabled in some way. Gas log fireplaces in many counties have to have disabled or clipped open dampers due to fire code. The reason for this is the county doesnt want you lighting a gas fireplace without opening the damper, since this can start a fire or asphexiate you and everyone in the home. Another possibility in this case could just be a worn out damper.
If you find that the cold air source is comming from a clipped open, disabled, or old and crusty damper and inexpensive insulation option is a chimney balloon. They can be bought on Northline Express or through the manufacturer at
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They are a inflatable plug that stops air passage. They can be used with a gas log since if you light a fire under them they melt and release (unlike a metal damper).
Good luck finding the cold air entry point, i know from experience that cold drafts like that can be uncomfortable.