From the rules:
From the point when the action is announced legally to the point where the individual cost is required, if payment of any of an action’s mandatory costs is interfered with, the player taking the action may choose to pay the costs another way, or choose not to pay the cost. ...
Sometimes an action, trait or card has a mandatory cost with a variable value, and this value depends on an element of the game that another player’s' effect interferes with at some point between announcement and the point when the cost is paid. Example: An action’s Gold cost depends on the total Gold cost of a target unit. After announcement but before costs are paid, a card is added to the unit or the targeting of the unit is changed. When the basis for a cost that depends on targeting changes due to interference with targeting, this is treated the same way as interference with the payment of the cost.
(Underlines and bold are mine) Just to clarify, if an action dependent on a target unit's gold cost had it's target changed from Target A to target B, but if Target A and Target B have the exact same gold cost, is this considered to be an "interference with the payment of the cost." It doesn't seem to be the case because (as far as I can tell) the rule is looking for the cost changing, not the target.