Hello Paul,
a new link up is a topology change as well.
In classical STP 802.1D this triggers a recalculation in the whole network.
At the end of the calculation the new port will become the new Root Port for its lower path cost to the Root Bridge.
Using rapid spanning-tree convergence time is faster and the synchronization concept will help to confine the changes. The two parts will perform an handshake on the new link about proposed roles and states and will update the state of non-edge ports (ports not in portfast) accordingly to new scenario.
Be aware that if some STP protection mechanism is configured the new link can even be placed in a spanning-tree inconsistent state as a result of its attempt to become root port (spanning tree root guard or loop guard)
Hope to help
Giuseppe