He was particularly strong on the horizontal bar, winning the 1971 and 1973 European championships and a bronze medal at the 1970 World championship in this event. Köste started training in gymnastics at the age of six in Frankfurt, but later moved to Leipzig where he lived for the rest of his life. He retired from competitions in 1974 due to an Achilles tendon injury and became a trainer and high school teacher.
In 1974–1976 he was the head coach of the East German women's team and between 1976 and 1985 worked as the chief trainer of South Carolina Leipzig.
In parallel he taught sports at DHfK Leipzig. He died from a heart failure in 2012.