The following figure shows the log-likelihood of individual observations for different subsets of a data set - the horizontal line shows the average log-likelihood:
So, the first subset includes all the data points observed up to t=7, the second all data points up to t=33, and so forth. For each subset, I estimate a parameter $\theta$, and then I calculate the likelihood of each observation given this parameter estimate. This is what the figure shows.
It can be seen that on average the likelihood of the observed data (shown as a horizontal line) tends to decrease as the number of observations grows. This pattern can be confirmed in the following plot, which shows how the average likelihood of observations changes as the number of observations included grows, for different data sets:
In the light of this, would you agree that these results suggest that the $\theta$ parameter is not equal for all $t$? If it was equal, I would expect the same average likelihood regardless of the number of observations included.