And since it happens after the car has been running for 10 mins, the t-stat is probably opening at that point and the old hose simply is lacking the structural integrity to maintain its wall shape.
If the hose is soft when the motor is cold but gets hard after the car has been running for a bit (or when the hose bulges), that suggests the t-stat is opening.
However, a new stat and bypass hose would be worth doing while swapping the hose. Other hoses might be in the same shape so they should be considered as well.