  We derive analytical upper bounds for the entanglement of generalized
Greenberger-Horne-Zeilinger states coupled to locally depolarizing and
dephasing environments, and for local thermal baths of arbitrary temperature.
These bounds apply for any convex quantifier of entanglement, and exponential
entanglement decay with the number of constituent particles is found. The
bounds are tight for depolarizing and dephasing channels. We also show that
randomly generated initial states tend to violate these bounds, and that this
discrepancy grows with the number of particles.
