  20 years ago, Bovier, Kurkova, and L\"owe [5] proved a central limit theorem
(CLT) for the fluctuations of the free energy in the p-spin version of the
Sherrington-Kirkpatrick model of spin glasses at high temperatures. In this
paper we improve their results in two ways. First, we extend the range of
temperatures to cover the entire regime where the quenched and annealed free
energies are known to coincide. Second, we identify the main source of the
fluctuations as a purely coupling dependent term, and we show a further CLT for
the deviation of the free energy around this random object.
