A blended composition of unsaturated block copolymer with improved thermal stability and processing behavior includes at least one unsaturated block copolymer; and a compatibilizer selected from the group consisting of (1) high melt flow rate homopolymers or copolymers; (2) styrene-ethylenepropylene-styrene (SEPS); (3) ethylene vinyl acetate (EVA); (4) styrene-butadiene-styrene (SBS), or styrene-isoprene-styrene (SIS) block copolymers; (5) single site catalyzed polyolefins, such as metallocene catalyzed and constrained geometry polyolefins; (6) amorphous poly alpha olefin homopolymer and copolymers; and (7) a combination of such.