  We address systematically an apparent non-physical behavior of the free
energy moment generating function for several instances of the logarithmically
correlated models: the Fractional Brownian Motion with Hurst index $H = 0$
(fBm0) (and its bridge version), a 1D model appearing in decaying Burgers
turbulence with log-correlated initial conditions, and finally, the
two-dimensional logREM introduced in [Cao et al., Phys.Rev.Lett.,118,090601]
based on the 2D Gaussian free field (GFF) with background charges and directly
related to the Liouville field theory. All these models share anomalously large
fluctuations of the associated free energy, with a variance proportional to the
log of the system size. We argue that a seemingly non-physical vanishing of the
moment generating function for some values of parameters is related to the
termination point transition (a.k.a pre-freezing). We study the associated
universal log corrections in the frozen phase, both for log-REMs and for the
standard REM, filling a gap in the literature. For the above mentioned
integrable instances of logREMs, we predict the non-trivial free energy
cumulants describing non-Gaussian fluctuations on the top of the Gaussian with
extensive variance. Some of the predictions are tested numerically.
