What temperature does is up to a GM, but Constitution checks to avoid exhaustion seem reasonable
The entire section on "Weather" (page 110) from the DMG only mentions a few weather types:
- Extreme Cold
- Extreme Heat
- Strong Wind
- Heavy Precipitation
Those are all it mentions, and the section on "Extreme Cold" (page 110) states:
Whenever the temperature is at or below 0 degrees Fahrenheit, a creature exposed to the cold must succeed on a DC 10 Constitution saving throw at the end of each hour or gain one level of exhaustion...
There is also a section on "Frigid Water" (page 110) which states:
A creature can be immersed in frigid water for a number of minutes equal to its Constitution score before suffering any ill effects. Each additional minute spent in frigid water requires the creature to succeed on a DC 10 Constitution saving throw or gain one level of exhaustion...
This is even more impacting that extreme cold - having you make a save every minute instead of every hour. Perhaps you could use this to say that cold temperatures cause saving throws every 2-3 hours.
This is also supported as the section on "Exhaustion" states:
Some special abilities and environmental hazards, such as starvation and the long-term effects of freezing or scorching temperatures, can lead to a special condition called exhaustion...
Unfortunately it never mechanically defines how to determine whether freezing temperatures result in exhaustion and "freezing temperatures" probably isn't
a literal term. Regardless exhaustion could very well be used with temperatures between 0° F (-17° C)? and 32° F (0° C), the GM would just have to determine a frequency of the Constitution saves and their DC.
The other place I found mention of temperatures is the control weather spell which states:
[...] When you change the weather conditions, find a current condition on the following tables and change its stage by one, up or down.
When you cast the spell, you change the current weather conditions, which are determined by the DM based on the climate and season.
- 1 Unbearable heat
- 2 Hot
- 3 Warm
- 4 Cool
- 5 Cold
- 6 Arctic cold
However, the spell never goes on to explain what any of these temperatures might do, nor does it mention any source or reference to determine what they do.
This seems to say that it would be left up the GM to determine the effects of weather, be they from this spell or elsewhere.