general relativity - Does the age of the universe depend on the way it is defined?

I was reading some earlier posts on this question in stack exchange. These posts give me the impression that the age of the universe is defined in a certain way following prescriptions such as that:

• 
  

  you should be co moving with the galaxy

  
  


• 
  

  avoid strong gravitational fields

  
  


• 
  

  choose the longest proper time recorded in all the frames

  
  
  

Defined this way we get the feeling that the age of the universe is not quite objective. Because we will get a different age if we change the definition. To highlight this arbitrariness, let us consider two space-time travellers A and B equipped with rockets and enough fuel who start at the same space point after setting their respective clocks to zero. Let us say that they started long time ago and took very different space-time paths and meet now and compare their clock readings T(A) and T(B). Each one can in principle claim that his clock showed the age of the universe. One may go a bit further and send a large number of travellers and collect all their clock readings at the end and declare the longest reading as the age of the universe. In this case we can say that the traveller who did not fire his rocket is the winner.

The implication in accepting this definition is that the time shown in this clock is the objective or absolute time of the universe while the times recorded in the other clocks are subjective. This way this definition will bring in a preferred reference frame and make the age of the universe an absolute time and not relative. In relativity theories we have not so far invoked preferred reference frames or absolute time concepts. Should the age of the universe be an exception?

I see that my post has some overlap with some earlier posts.However the important difference is that, as agreed by all who responded to my post,the earlier posts did not indicate that the age of the universe could be as arbitrary as the choice of a reference frame and no reference frame has any privileged status compared to the others.If any one wants to use a particular frame for some reason or another we can not question him.But one can not and should not object to the use of a different frame which computes a much younger universe, say an universe only 2000 years old !!! Lo and Behold! 2000 years can not be true as we know that we existed from much earlier days onwards. Hence the interesting question is: what may be a more realistic lower bound for the age of the universe?