Tips of aloe plant
Thank you for including images with your question. It appears that the leaf tip in IMG_0545 is slightly off-color. If it is also soft, your thought of cold damage is accurate.
Aloe is a subtropical plant. Thus, it can be adversely affected by “chilling damage” which occurs when temperatures are above 32 degrees Fahrenheit but below 59 degrees Fahrenheit. See Chilling Injury, Frost and Freeze Injury: What's the Difference?
Another publication, Growing Indoor Plants with Success (page 17), describes the temperature needs for indoor aloe plants to be “70°F night, 85°F days.” So, even if the leaf tips aren’t yet soft or rotting, it would be wise to move the plant away from the window, especially during the coldest hours.