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.
Montreux Jazz Festival was created in 1967.
Charlie Jazz Festival was created in 1998.
Berkeley Jazz Festival was created in 1967.
Brecon Jazz Festival was created in 1984.
Hampton Jazz Festival was created in 1968.
Newport Jazz Festival was created in 1954.
Umeå Jazz Festival was created in 1968.
Toronto Jazz Festival was created in 1987.
Cascais Jazz Festival ended in 1988.
Cascais Jazz Festival was created in 1971.
Nice Jazz Festival was created in 1948.
Montreux Jazz Festival was created in 1967.
Cork Jazz Festival was created in 1978.
Tórshavn Jazz Festival was created in 1983.
Koktebel Jazz Festival was created in 2003.
Concord Jazz Festival was created in 1969.
Rome Jazz Festival was created in 1976.