  We study the logarithmic entanglement negativity of symmetry-protected
topological (SPT) phases and quantum critical points (QCPs) of one-dimensional
noninteracting fermions at finite temperatures. In particular, we consider a
free fermion model that realizes not only quantum phase transitions between
gapped phases but also an exotic topological phase transition between quantum
critical states in the form of the fermionic Lifshitz transition. The bipartite
entanglement negativity between adjacent fermion blocks reveals the crossover
boundary of the quantum critical fan near the QCP between two gapped phases.
Along the critical phase boundary between the gapped phases, the sudden
decrease in the entanglement negativity signals the fermionic Lifshitz
transition responsible for the change in the topological nature of the QCPs. In
addition, the tripartite entanglement negativity between spatially separated
fermion blocks counts the number of topologically protected boundary modes for
both SPT phases and topologically nontrivial QCPs at zero temperature. However,
the long-distance entanglement between the boundary modes vanishes at finite
temperatures due to the instability of SPTs, the phases themselves.
