First increase the damage, then apply reductions
The developers have explained how those abilities should work, based creatures that appear on certain adventure paths and monsters that do have that combination of abilities on the bestiaries.
In Mummy Mask #1 (The Half-dead City), there is a clay creature that has a vulnerability to fire, and being a construct, they have hardness , but first you increase the damage by 50%, then you apply hardness, then anything that goes through is normal damage, as explained by the author of the adventure and later confirmed by Robert G. McCreary (a developer).
The animated objects are constructs, so they are now creatures, not objects. As creatures, energy damage is not halved against them (in effect, becoming a creature trumps the normal object rules). So they take full damage from energy attacks (150% if they are vulnerable to that energy type), then hardness is applied. However, page 174 of the Core Rulebook states (under "Vulnerability to Certain Attacks") that "Certain attacks are especially successful against some objects.In such cases, attacks deal double their normal damage and my ignore the object's hardness."
Even though the animated objects are no longer "objects," I would still apply this rule about overcoming hardness to them in this adventure (but the 150% creature vulnerability trumps the double damage to an object part of the rule), as hardness can be quite difficult to overcome for low-level PCs. So if the PCs use fire against the warrior dolls, for example, it would deal 150% the normal damage, and ignore the hardness. Other energy damage would deal full damage which would ten be reduced by hardness, the same as for any other attack against it.
Another example are the robot creatures from the Iron Gods adventure path, where they have linked to McCreary's answer several times when questioning the order of abilities and again confirmed by James Jacobs on his unofficial thread, because all robots are vulnerable to electricity, but some of them are also resistant to electricity, like the iconic Annihilator from the Inner Sea Bestiary cover, and even though they have hardness, they should not be treated as objects, just like other constructs.
You need to determine the total amount of damage the robot WOULD take before applying hardness and electricity resistance. So once the damage is determined, first thing you do is apply the vulnerability effect to get the total damage. Hardness and resistance kick in after that.