According to Gregory, the exposure to the long party of a credit default swap increases in its early years and then skyrockets when there is a credit event of the reference entity.

I would have suspected the exposure starts off negative and then skyrockets when there is a credit event.

Is this behaviour related to the theory that default probabilities increase with time?

In case I misunderstood (I don't think I did), here is the source text

Consider the exposure profile of a single-name CDS as shown in Figure 8.20 (long CDS protection). The exposure increases in the early stages, which corresponds to scenarios in which the CDS premium (credit spread) will have widened. However, the maximum exposure on the CDS corresponds to the reference entity experiencing a credit event, which triggers an immediate payment of the notional less a recovery value