Natural rubber is made of polyisoprene chains that slip past each other when the material is stretched.
When raw, the substance is too sticky and soft to be of much use, so it is toughened with the addition of chemicals such as sulphur that create cross-links between the chains, making the rubber stiffer and less sticky.
This process is called vulcanization.
With time, ultraviolet light and oxygen in the air react with the rubber, creating reactive radicals that snip the polyisoprene chains into shorter segments.
This returns the rubber to something like its original state, soft and sticky.
Meanwhile, these radicals can also form new, short cross-links between chains. This hardens the rubber and eventually it turns brittle. Any vulcanization agents left in the rubber contribute to the process.
Whether a rubber band goes sticky or hard depends on the relative rates of these processes, and these rates in turn depend on the rubber’s quality such as what additives, fillers, and dyes it contains, and how it is stored.
Heat and light speed up the reactions, for example, an 18°F rise in temperature will roughly double reaction rates, and the presence of strong oxidizers such as ozone creates even more radicals.
The eventual fate of your rubber band depends on the temperature in the room, and on whether you have a desk by a window or near a machine such as a photocopier that creates ozone.
How much light and heat is required for these changes?
The polymer chemistry of rubber is fairly messy, and so this is difficult to answer precisely. Obviously, the chemical reactions run slowly if the rubber is in a refrigerator, more quickly if left on a sunny desktop.
A rule of thumb is that reaction rates roughly double for an 18°F rise in temperature, but this is complicated when you take oxygen and light into consideration.
The quality of the rubber is also important, such as whether it contains additives, fillers, or dyes that absorb light energy or help transfer radicals.
The final factors that influence the change are ozone concentration, UV light intensity, and whether the band is stretched or not, stretching brings chains closer together, allowing radicals to jump from one chain to another more easily, and to create new bonds between chains.