Mr Griffith-Jones had a hearing with the Treasury Select Committee last November on his FCA appointment.
He was warned of the failures of the FSA and told that the FCA should not follow the FSA's mistakes.
Mr Griffith-Jones was announced as the chair of the FCA in June 2012, he was previously chairman of KPMG.
The committee recommended that the FCA should follow the Prudential Regulation Authority's lead, conduct internal reviews and set up an effective Board.
Andrew Tyrie MP, chairman of the Treasury Select Committee, said: "It [the FCA] is taking over from a body, the FSA, which failed consumers badly. If the FCA simply picks up where the FCA left off, consumers will suffer again.
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"Regulation is primarily about judgement. Mr Griffith-Jones and his board will need to ensure that the FCA follows the Prudential Regulation Authority's lead and adopts a radically different approach to the widely discredited box-ticking practices undertaken by the FSA."
Mr Tyrie also brought attention to the FCA's objectives which he felt could be conflicting priorities.
"The Treasury Committee has expressed deep misgivings about the conflicting priorities inherent in the FCA's hierarchical objectives. Not least among its problems is that they may lead to inadequate attention being paid to the competition objective. Mr Griffith-Jones will need to think very hard about the inter-relationship between these objectives and how they can be meaningfully fulfilled."

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