  It has been argued that the quantum (conformal) trace anomaly could
potentially provide us with a dynamical explanation of the cosmological
constant problem. In this paper, however, we show by means of a semiclassical
analysis that the trace anomaly does not affect the cosmological constant. We
construct the effective action of the conformal anomaly for flat FLRW
spacetimes consisting of local quadratic geometric curvature invariants.
Counterterms are thus expected to influence the numerical value of the
coefficients in the trace anomaly and we must therefore allow these parameters
to vary. We calculate the evolution of the Hubble parameter in quasi de Sitter
spacetime, where we restrict our Hubble parameter to vary slowly in time, and
in FLRW spacetimes. We show dynamically that a Universe consisting of matter
with a constant equation of state, a cosmological constant and the quantum
trace anomaly evolves either to the classical de Sitter attractor or to a
quantum trace anomaly driven one. When considering the trace anomaly truncated
to quasi de Sitter spacetime, we find a region in parameter space where the
quantum attractor destabilises. When considering the exact expression of the
trace anomaly, a stability analysis shows that whenever the trace anomaly
driven attractor is stable, the classical de Sitter attractor is unstable, and
vice versa. Semiclassically, the trace anomaly does not affect the classical
late time de Sitter attractor and hence it does not solve the cosmological
constant problem.
