Gaussian process regression: sudden increase of the prediction's variance
Hmm, I've seen this before. It seemed to be some kind of overfitting on the time component where only very nearby samples were taken into account, before defaulting to the prior (which has zero mean). That would explain why increasing the length scale would help. 

Do you see large changes in the hyper parameters of the kernel between iterations (gp.kernel_.get_params())? You could consider using the hyper parameters from a previous iteration as initial guess for the next iteration.