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.
Hampton Jazz Festival was created in 1968.
Cotati Jazz Festival was created in 1980.
Montreux Jazz Festival was created in 1967.
Berkeley Jazz Festival was created in 1967.
Brecon Jazz Festival was created in 1984.
Hampton Jazz Festival was created in 1968.
Skopje Jazz Festival was created in 1982.
Beijing Jazz Festival was created in 1993.
Cotati Jazz Festival was created in 1980.
Saulkrasti Jazz Festival was created in 1997.
Chicago Jazz Festival was created in 1978.
Gărâna Jazz Festival was created in 1997.
Rome Jazz Festival was created in 1976.
Malta Jazz Festival was created in 1990.
Newport Jazz Festival was created in 1954.
Umeå Jazz Festival was created in 1968.
Toronto Jazz Festival was created in 1987.