I work on the principle that you have a goal, singular, and that everything you have done, or are currently doing, or are going to be doing, is preparation for achieving that goal. this goal usually lies some years in the future. individual seasons might have targets, that are stepping stones along the way to your goal, but they are not of themselves what drives or motivates your preparation or planning.
coaching is merely choosing the specific stress that induces the specific adaptation that makes your athlete better suited to meet their performance goal.
if your goal is to win the olympics, for example, then you will at some point have to peak twice in a season. first time to win the trials, then second time around to win the goal race some six months later.
I would not have thought that you would want the first time you do this to be mission critical. you want to have prepared for this so that you know what happens in those six months, recovery-wise, and bounce-back-wise.
you will also want to know how to handle three rounds of heats, semi-final and final in five days, so you will have prepared for that, too.
so the most obvious pro of double periodisation is that it prepares the athlete for what happens in real life competitive situations rather than in text book coaching scenarios.
the most obvious con of double periodisation, as far as I can tell, is that for the 99.9% of athletes who are never going to make the trials much less the olympics, it is an unnecessary stress on the system that does nothing towards achieving their goal.
you don't NEED double periodisation to win the NCAA, or to break 27:00 for 10k, or to triple jump 17 metres or whatever your goal is. you only need it if, ironically, you actually need it.
in technical events, high jump, for example, we do sometimes compete indoors off the new strength developed in the previous strength phase where we are looking for an expression of that strength but not overly concerned with technique. then in the spring we work a lot on technique to refine the strength gains and prepare for track so we kind of get one and a half peaks where we are looking for different aspects of performance. in that case the indoor is a kind of wind down and fun break from consistent hard work, it isn't really a double peak.
for a runner, this might equate to racing under distance indoors as preparation for a faster pace in the work we are going to be doing in the spring. but I would still consider this to be training rather than peaking, it's a kind of look-see to find out where we are rather than an actual peak.
it all depends on what your goal is.
cheers.