The problem with this approach is TTFB (Time to first Byte).  The
initial query is very slow, but additional requests are fast.  In most
situations we do not want the user to have to wait a disproportionate
amount of time for the initial query.  If this is the first time using
the system this will be the impression that will stick with them.  I
guess we could experiment and see how much extra time creating a cache
table will take...

Do it on the second page then ;)

Seriously :
- If you want to display the result count and page count, you'll need to do the whole query anyway, so you might as well save the results.
- inserting the result id's in a temp table one by one will be slow, but you can do this :

select array_accum(id) from temp group by id/20 limit 3;

- a really big search of 131072 results :
create table cache (id serial primary key, value integer[]);
explain analyze insert into cache (value) select array_accum(id) from temp group by id/100;
Subquery Scan "*SELECT*" (cost=14382.02..17986.50 rows=131072 width=32) (actual time=961.746..1446.630 rows=1311 loops=1)
-> GroupAggregate (cost=14382.02..16020.42 rows=131072 width=4) (actual time=961.607..1423.803 rows=1311 loops=1)
-> Sort (cost=14382.02..14709.70 rows=131072 width=4) (actual time=961.181..1077.662 rows=131072 loops=1)
Sort Key: (id / 100)
-> Seq Scan on "temp" (cost=0.00..2216.40 rows=131072 width=4) (actual time=0.032..291.652 rows=131072 loops=1)
Total runtime: 1493.304 ms

Note that the "SELECT..." part takes 1400 ms, and the INSERT part takes the rest, which is really small. It's the sort which takes most of the time, but you'll be doing it anyway to get your results in order, so it comes free to you. This will generate 1000 pages with 100 results on each. If your searches yield say 1000 results it'll be perfectly fine and can target times in the sub-100 ms for caching the results (not counting the total query time of course !)

Using arrays is the key here, because inserting all the results as individual rows in the table is gonna be a whole lot slower !

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