I have placed asphalt concrete in slightly rainy weather with no impact to quality.  Asphalt is porous enough, that if it is properly compacted, a little moisture in the mix during placement should not impact it significantly.

Water will make it cool fast, and therefore the "breakdown" roller will have to be applied sooner

Placement under the conditions you described (particularly paving over birdbaths with standing water) will result in a lack of bond between the courses of asphalt.  In the short term, you will likely see some "blistering" of the upper course, followed by a fracture around the blister, then localized loss of the layer.

I agree with DSOLO that the breakdown rolling would have to start sooner.  The window of opportunity for proper compaction is smaller under adverse temperature where the asphalt cools quickly.  If not done properly, this will result in an undercompacted mix that will lead to raveling and/or surface compaction rutting with time (as the traffic compacts the undercompacted layer).

Undercompaction also increases the voids in the asphalt which, in turn, allows water intrusion into the layers.  While some mixes are designed to accomodate "subsurface" drainage, those that are not will likely come apart under those conditions.

Asphalt which is applied at a lower than specified temperature will deteriorate over time.  Check temperatures of mix brought to project.  Note and keep record of questionable areas.