Let us denote our credence in proposition P at time t as follows: Ct(P).
At t=100, we have C100(P) = 0.7. Then we wait a very very long time. I don't want to write as many zeros as you wrote, so let's just say that after this very long time, the time is t=500. The question is, what should C500(P) be?
First, let's agree (as is extremely common) that credences are updated via Bayesian Conditionalization. That is, C500(P) = C100(P|E), where E denotes all of the evidence we obtain between t=100 and t=500. The idea is that we ask, at t=100: what credence would we have if we happened to learn E? That's C100(P|E). Then we actually observe E. Bayesian Conditionalization says: our new credence C500(P) should be our old conditional credence C100(P|E). And our old conditional credence can be calculated by Bayes' Theorem: C100(P|E) = C100(E|P)C100(P)/C100(E).
You ask us to suppose that we "gather no relevant information about P after t100." Let's take an extreme example of this. Suppose I sit in a black box, staring blankly, until t=500. What evidence have I gathered? Nothing very interesting, but I have gathered some evidence. I've learned the truth of the proposition "I have continued to exist from t=100 to t=500." Now, depending on what P is, this could be relevant information. For example, if P is the hypothesis "I am immortal," and if the time from t=100 to t=500 is ridiculously long for members of my species, presumably my credence in P should go up at t=500. Living for an absurdly long time is good evidence for immortality. But I assume this is not the kind of case you have in mind.
So suppose P is instead the hypothesis "All ravens are black." The fact that I survived from t=100 to t=500 is, intuitively, not relevant information for this hypothesis. And in a probabilistic setting like this, irrelevance entails statistical independence. Let E be all the evidence I learned in my black box. Since E is irrelevant to P, they are statistically independent, so (by definition of independence) C100(E|P) = C100(E). You can check that, given what I said above, it follows that C500(P) = 0.7. My credences haven't changed.
However, you also suppose that we were "rigorously looking for evidence to (dis)confirm P." Clearly, sitting in a black box doesn't count as rigorously looking for evidence! So suppose we rigorously look for evidence, but still find nothing that we deem relevant to P. In that case, even though we looked and looked, the argument I gave in the previous paragraph remains, and our credences remain unchanged: C500(P) = 0.7.
Now, let's imagine one final case. Suppose our hypothesis P is "Ravens are fairly common on Earth." Then we travel around the globe frantically, looking for ravens, but find nothing. We find no obvious evidence of their presence. We don't find abandoned ravens' nests, or fresh raven skeletons. And we also don't find any obvious evidence that (e.g.) they were all made extinct by some environmental catastrophe. We just looked at a bunch of places, and found nothing obviously raven-related.
In this case, it's tempting to say that we acquired no evidence relevant to P. But in fact we did! We learned the proposition E, namely "I didn't find any ravens, even though I looked long and hard." Now, let's suppose that at t=100 (before we gather this evidence), our credence that we will observe that evidence is low. Let's say C100(E) = 0.5. But notice that, if our hypothesis P ("Ravens are very common on Earth") is true, then we expect the evidence to be very unlikely. For example, let's say C100(E|P) = 0.01. This shows that E and P are not statistically independent. Hence, E cannot be irrelevant to P. But indeed, Bayesian conditionalization tells us that our credence at t=500 should be much less than 0.7.
The moral of the story: if your evidence E is truly irrelevant to P, then observing it cannot change your credences in P.
