Robert posted a possible cause. If this is the cause, then you have to consider future repair even though the PC boots-up until it will eventually fail.
Otherwise, you can make it better by providing some "direct heat" to the PC. the use of an electric heater for a few mins. before booting or heat the room. You could also place the PC in a "hot box" or boxy cover with a light bulb to keep the temp. up before use during these colder months. Then remove when good weather arrives. This is used in garages and anywhere usually some outside location, etc..
This all maybe a sign of it showing its age. Due to wear&tear, the HD or components themselves just don't cut it. After-all, there are "environmental concerns" to make a minimum operating area. What maybe 40deg. to you maybe 35deg. to the PC location and/or metal areas, etc.. all explained in the specs. As usual a good cleaning is recommended as well to make sure "dust bunnies" aren't causes some issue, reducing the fans, yada, yada a strain on the PC to even boot-up. Which all could lead a PSU starting to fail, again due to wear&tear.
My web client's computer takes anywhere from 10 to 20 minutes to start-up when his office temperature is below 40 degrees in the morning. When his office is warm, around 70 degrees, the computer start-up time is about 2 minutes.
Is it time to replace the computer, leave some heat on in the office, or is there something that can be done to the computer so it can boot up in 2 minutes (as it does when the office is warm) from a COLD office environment?