  We study a topologically exact, negative Schwarzian unimodal map whose
critical point is non-recurrent and flat. Assuming the critical order is either
logarithmic or polynomial, we establish the Large Deviation Principle and give
a partial description of the zeros of the corresponding rate functions. We
apply our main results to a certain parametrized family of unimodal maps in the
same topological conjugacy class, and give a complete description of the zeros
of the rate functions. We observe a qualitative change at a transition
parameter, and show that the sets of zeros depend continuously on the parameter
even at the transition.
