The Abbey of Anomalies is a special District that can only be built once per city and must be built on an Anomaly. Doing so will double the bonuses that Anomaly provides to Food, Industry, Science, Dust, Influence, and Approval. This effect also applies to any bonuses that don't come from the Anomaly itself, such as base terrain bonuses, City Improvement, Hero skills, or temporary world effects.
The Abbey also provides an additional +5 Food per level to the Anomaly it is placed on (which isn't doubled by the previous effect), as well as increasing the Food output of all other anomalies in the region by 3 .
While the Abbey of Anomalies is available to all factions, the Broken Lords do not receive any benefit from the Food effects of the Abbey, and The Forgotten do not receive any benefit from the Science effects of the Abbey.
It is a place rich in the powers of Dust and Auriga's Pearls, and sanctified by prayer. Diligent study and work at the Abbey enhance the ability to draw riches from Anomalies in the city.
Because the greatest benefit from the Abbey of Anomalies is to double the FIDSI and Approval from Anomalies, it is important to build it on an Anomaly that provides a high total output. In general it is best to build the Abbey on an Anomaly that provides a boost to Approval, not only to gain the extra benefit from the increased Approval, but also because anomalies that provide a bonus to Approval also have a higher total FIDSI output; Anomalies with no Approval bonus have a total FIDSI output of 6, while those with an Approval bonus of 10 have a total FIDSI output of 8, and the Golden Tree Anomaly - which has an Approval bonus of 20 - has a total FIDSI output of 10 (2, 8).
That said, it is also important to consider other factors, such as which factions you're playing as: for example, the Broken Lords would be better off placing the Abbey on the Ruby Cactus Fields Anomaly (2, 4) than they would from placing it on the Oasis Anomaly (6, 2, 10) since their inability to use Food reduces the otherwise superior Anomaly's FIDSI output from 8 to 2. The same reasoning applies to The Forgotten.
However, one of the benefits of the Abbey of Anomalies is that it is easy to plan its placement far in advance. Because all anomalies are revealed from the start of the game, you can pick an anomaly near your city to build towards and/or save for when you unlock the Abbey to build it there.