Car horn switches are not hermetically sealed or moisture free (unlike some relays found in cars - experience). Assuming your switch/button horn activator had previously fallen below dew point and moisture had condensed on it's contact surfaces and subsequently frozen - ice is not a good conductor of electricity and thus the horn didn't work. Horn switches are normally of the direct push type pressing contacts together on one axis.
Indicators (turn signal) switches are normally of the sliding sort and thus to some extent self cleaning, this is also true of the Tail and headlight switch in my car, perhaps in yours too, thus they are more likely to work first time.
As an experiment, the next time the morning is frosty, you could try pressing the horn repeatedly and hard, this should transfer enough heat energy to de-frost on the contacts and make it work. If this is not effective then an assumption about the diaphragm of the horn being less flexible at frosty temperatures, or the voltage supplied not being sufficient - because you had just started the car (using more than a few amps, starter motors are very power hungry), turned on the lights, rear window demister circuit etc. and the voltage in the battery/alternator circuit had not recovered to one sufficient to make it work.
These are all speculative to some extent, without making specific measurements regarding resistances and voltages, and moisture levels the previous day and of course temperature; it is difficult to do anything other than speculate.
