  We investigate further the relationship between the entanglement spectrum of
a composite many-body system and the energy spectrum of a subsystem making use
of concepts of canonical thermodynamics. In many important cases the
entanglement Hamiltonian is, in the limit of strong coupling between
subsystems, proportional to the energy Hamiltonian of the subsystem. The
proportionality factor is an appropriately defined coupling parameter,
suggesting to interpret the latter as a inverse temperature. We identify a
condition on the entanglement Hamiltonian which rigorously guarantees this
interpretation to hold and removes any ambiguity in the definition of the
entanglement Hamiltonian regarding contributions proportional to the unit
operator. Illustrations of our findings are provided by spin ladders of
arbitrary spin length, and by bilayer quantum Hall systems at total filling
factor nu=2. Within mean-field description, the latter system realizes an
entanglement spectrum of free fermions with just two levels of equal modulus
where the analogies to canonical thermodynamics are particularly close.
