2010 Accord no crank no start
 

My 2010 Accord has an interesting and annoying problem. Whenever the weather becomes cold, it won't start. Sometimes it took several tries to get the car started. Yesterday it refused to start at all. Usually one would expect that a bad battery causes the problem, but mine seems to be perfectly fine with a 12.4 V.
I suspect that this may be starter related but not sure whether this is caused by a bad starter because it seems to be temperature-dependent. Could this be due to a bad connection or relay?