Any fire with flames is something around 600C and up to even double that. Air is relatively cheap to heat up. I'd wager the combustion process, once fully started, won't suffer much if the ambient temperature is -30C or +30C: the fire is still on a temperature scale that's an order of magnitude higher.

Even on a summer day a good breeze of wind or just blowing into the fire too hard yourself will put it out but only if it was just starting. Once the fire is rooted in something more solid, combustible material it will easily heat up any fresh air that is conveyed into the fire.

Air is cheap to warm.

Water in fuel is expensive

Though if this is a peat fire the water in the peat may have partially evacuated during the freezing process - I'm not so certain that the volume needing to be sent from ice -> water -> vapor would take more energy than the larger volume sent from water -> vapor... I really have no knowledge of ratios here but there are at least some processes working against increasing the amount of energy that needs to be expended to heat the surrounds.

Additionally, if this fire is mostly underground then it's likely that you've got some oven action going on where a lot of the heat produced by the fire isn't just whisked away by air to dissipate to nothingness - instead it's trapped by the insulation of earth and being converted into phase changes more efficiently.

> a lot more energy

Not relatively. The difference is still a small fraction of the total energy needed even on the hottest day.