I started on June 7, 2015 arriving in Santiago 0n July 23. It was hot but I acclimatised easily coming from Darwin. The heat is like Central Australia with that dry heat where the sweat evaporates off you quickly. Parts of the Sanabres remind me of Central Otago in NZ. There was a heat wave that year but I seemed to find the one cloud in Spain on quite a bit of my trip so it was cloudy and on one occasion, drizzly. But it was also stinking hot especially on stretches when there are big road stretches. Northern Extremadura is the worst in that respect and it is really important to get your stops and distances right otherwise you will cop short or very long distances. You cannot take too much water and it is really important to manage water well otherwise you will be more prone to blisters. I was averaging about 250 ml per hour of water and would also take every opportunity to drink. I found that I felt cooler wearing sunnies and not having a hat because my dark hair traps heat. Since then I use a visor when hiking as shade over my eyes makes my brain think it is cooler. I also carry a silver reflective umbrella and it is like carrying your own portable shade treat and takes the temp down a couple of degrees which can save your arse on horror days.
Heat index is worth a read in that respect. Well worth being out the door at dawn so that you have finished for the day by 3 pm at the latest as that is the hottest time of the day. I try to avoid walking when it is hotter than body temp. I tend to average 4 km per hour which includes toilet stops, breaks and taking lots of photos. Cafe con hielo becomes your favourite form of coffee. The good thing is that you are very unlikely to need heavy gear so you can travel much lighter compared to Feb/Mar when you have to consider late winter storms (been there, done that).
https://weatherspark.com/ is a great site because you can work out the range of likely conditions for where you will be based on historical records.
I did a Live from the Camino thread which gives a bit of detail. Everything is very golden in June and July and I find photos of a green VDLP quite weird. Summer is definitely doable with caution and good planning.