I feel the performance characteristics change from one DBMS to another. It's all on how they choose to implement it. Since I have worked extensively on Oracle, I'll tell from that perspective.
COUNT(*)
- Fetches entire row into result set before passing on to the count function, count function will aggregate 1 if the row is not null
COUNT(1)
- Will not fetch any row, instead count is called with a constant value of 1 for each row in the table when the WHERE
matches.
COUNT(PK)
- The PK in Oracle is indexed. This means Oracle has to read only the index. Normally one row in the index B+ tree is many times smaller than the actual row. So considering the disk IOPS rate, Oracle can fetch many times more rows from Index with a single block transfer as compared to entire row. This leads to higher throughput of the query.
From this you can see the first count is the slowest and the last count is the fastest in Oracle.