This logic seemed obviously to me. Why shouldn’t dying suns have a window of time in which life can arise on a frozen planet? Global warming makes this obvious, in that arctic regions become more hospitable. The 1 billion year frame gives plenty of time to get life started. What might really complicate this is if it turns out that the proximity of a sun is not all that important in the triggering of life, which may turn out to be true on Titan. If there was primitive life on a planet or moon like Titan in a system with a sun in the process of turning into red giant, would the complexity of the evolving organisms necessarily accelerate?