DDL

create table count_cache (
  id serial primary key,
  table_name varchar(255) not null,
  count int not null
)

create table people (
  id serial primary key,
  name varchar(255) not null
)

Triggers

-- insert trigger
create or replace function log_count_insert() returns trigger as $end$
  begin
    update count_cache set count = count + 1 where table_name = TG_RELNAME;
    return null; -- no return on an after trigger
  end;
$end$ language plpgsql;

create trigger people_insert_count after insert on people
  for each row execute procedure log_count_insert();

alter table people enable trigger people_insert_count;


-- delete trigger
create or replace function log_count_delete() returns trigger as $end$
  begin
    update count_cache set count = count - 1 where table_name = TG_RELNAME;
    return null; -- no return on after trigger
  end;
$end$ language plpgsql;

create trigger people_delete_count after delete on people
  for each row execute procedure log_count_delete();

alter table people enable trigger people_delete_count;

--initialize on count_cache, this must be done for each table that caches the count
insert into count_cache(table_name, count) values ('people', 0)

Try it out

insert into people (name) values ('john');
insert into people (name) values ('john');
insert into people (name) values ('jay');
insert into people (name) values ('mustang');
insert into people (name) values ('sally');
insert into people (name) values ('fred');

select count from count_cache where table_name = 'people';
select count(*) from people;
--  => 6

delete from people where name = 'john';

select count from count_cache where table_name = 'people'
select count(*) from people;
--  => 4