The pressure inside a basketball decreases as the temperature decreases.
The “bounce” in a basketball depends on its having a certain internal pressure.
When a basketball hits the gym floor, the force of the impact pushes the bottom surface of the ball in and compresses the gas inside.
According to , the pressure of the gas inside increases. The increased pressure causes the ball to expand and bounce into the air.
On a cold day, the pressure inside the basketball decreases (Gay-Lussac’s Law).
Since the pressure of the gas inside the basketball is lower, the impact with the ground has less effect on the restoring force. The basketball does not bounce as high as it does in a warm room.