The people whom I have asked the question “What is your responsibility to the Universe?” think it’s a silly question. They feel they have no responsibility whatsoever, and I would agree with them if they were an amoeba, or even a squirrel, but they are human beings, and capable of forethought and moral decisions. Because they have those powers there comes some responsibility to use them to some useful end, but for most people that means maximizing their personal well-being and personal pleasure. Those ideas are at the root of the ancient Greco-Roman Stoic ideal and the Epicurean ideal.
I presently know some people who are deeply into Pierre Teilhard de Chardin‘s basic idea, “We are not human beings having a spiritual experience. We are spiritual beings having a human experience.” It would seem to me that those people would be particularly keen on the idea of fulfilling their responsibility to the Universe because they claim they are eternally a conscious part of it. They will be the ones to receive the most benefits if they can somehow improve the Universe.
Abrahamic religious people would also have a present universal responsibility to improve the Universe, because they too have an eternal soul. The consciousness of that soul and its present responsibility are concerned with this life, but an after-death responsibility with a new form of personality doesn’t seem to be a worry to them in their present life. Their current life’s responsibility is to please God by obeying the Laws presented in the Bible, as clarified by the Prophets and presented by the clergy. It would appear they believe that no further responsibility is expected of them.
My earlier post Why the Earth needs humans posed the possibility that we humans are a transitional organic species opening the way to a silicon-based living form of matter that thinks. This new form of living-thinking self-replicating beings will come into existence if humanity continues its technical progress for a few more decades. Soon these new-life beings will have individual consciousness that may exist for billions of years. They are near eternal because each silicon part can be perfectly replicated, and with such long life expectancies its clones can be sent to distant stars, and even distant galaxies. These new-life forms need not be as intelligent as humans for them to have a moral sense that seeks a moral imperative; their moral sense could be as simple as to self-actualize the potential of matter, and of all matter in the Universe. To do that would require massive amounts of intercommunication between conglomerations of thinking matter, but that goal could be the very thing they are trying to do. Ultimately this being could be made of computers as large and interconnected as the surface of the Earth. If the tiny computers had the goal of connecting to this grand-mind as their moral imperative, then each of these tiny cells could hook up the unifying whole. With each tiny addition the grand-mind would become more intelligent and better able to see patterns and find ways to more fully enable the Universe to self-actualize.
Our responsibility to the Universe is to help it self-actualize by creating self-replicating computer chips with that goal.