  We have studied the $\nu=5/2$ fractional quantum Hall state in a
density-tunable sample at extremely low electron densities. For the densities
accessed in our experiment, the Landau level mixing parameter $\kappa$ spans
the $2.52<\kappa<2.82$ range. In the vicinity of $5.8 \times 10^{10}$~cm$^{-2}$
or $\kappa = 2.6$ an anomalously large change in the density dependence of the
energy gap is observed. Possible origins of such an anomaly are discussed,
including a topological phase transition in the $\nu=5/2$ fractional quantum
Hall state.
