  In this paper, we investigate nonlinearly charged AdS black holes in
four-dimensional critical gravity and study more exact black hole
thermodynamics under the effect of small statistical fluctuations. We compute
the correction to the thermodynamics of nonlinearly charged AdS black hole up
to the leading order. We discuss the stability of black holes under the
circumstances of fluctuation and find that fluctuation causes instability in
the black holes. Moreover, both the isothermal and adiabatic compressibilities
are also derived. Finally, we estimate the role of small fluctuations on the
equation of states and study the $P-v$ diagram of nonlinearly charged AdS black
hole.
