  We obtain a exponential large deviation upper bound for continuous
observables on suspension semiflows over a non-uniformly expanding base
transformation with non-flat singularities or criticalities, where the roof
function defining the suspension behaves like the logarithm of the distance to
the singular/critical set of the base map. That is, given a continuous function
we consider its space average with respect to a physical measure and compare
this with the time averages along orbits of the semiflow, showing that the
Lebesgue measure of the set of points whose time averages stay away from the
space average tends to zero exponentially fast as time goes to infinity.
Suspension semiflows model the dynamics of flows admitting cross-sections,
where the dynamics of the base is given by the Poincar\'e return map and the
roof function is the return time to the cross-section. The results are
applicable in particular to semiflows modeling the geometric Lorenz attractors
and the Lorenz flow, as well as other semiflows with multidimensional
non-uniformly expanding base with non-flat singularities and/or criticalities
under slow recurrence rate conditions to this singular/critical set.
