I don't know of any jazz festivals which go by the name 'Kool'. If you were referring to 'cool' jazz festivals, ie the most popular ones, then you'll generally find that they run in the early summer to late autumn when the weather is nicest. They do run throughout the year as both the northern and Southern Hemisphere have many jazz festivals.
That would be the Montreal International Jazz Festival. It currently holds the Guinness World Record as the worlds largest jazz festival. Around 3000 artists perform each year in over 700 concerts. In 2014 the festival is going into it's 34th year and tipped to be the biggest yet, so it's certain to hold on to its title!
For more info on the festival, see the related links section below.
Some of the most popular music festivals around the world for Jazz Music lovers are the New Orleans Jazz and Heritage Festival, Montreux Jazz Festival and Copenhagen Jazz Festival. There is also the Monterey Jazz Festival and the TD Toronto Jazz Festival.
Cotati Jazz Festival was created in 1980.
Brecon Jazz Festival was created in 1984.
Berkeley Jazz Festival was created in 1967.
Montreux Jazz Festival was created in 1967.
Charlie Jazz Festival was created in 1998.
Cotati Jazz Festival was created in 1980.
Brecon Jazz Festival was created in 1984.
Berkeley Jazz Festival was created in 1967.
Bangkok Jazz Festival was created in 2003.
Viersen Jazz Festival was created in 1969.
Bray Jazz Festival was created in 2000.
Cheltenham Jazz Festival was created in 1996.
Umeå Jazz Festival was created in 1968.
Cascais Jazz Festival ended in 1988.
Cork Jazz Festival was created in 1978.
Sarajevo Jazz Festival was created in 1997.
Beijing Jazz Festival was created in 1993.