  A key objective in nuclear and high-energy physics is to describe
nonequilibrium dynamics of matter, e.g., in the early universe and in particle
colliders, starting from the Standard Model. Classical-computing methods, via
the framework of lattice gauge theory, have experienced limited success in this
mission. Quantum simulation of lattice gauge theories holds promise for
overcoming computational limitations. Because of local constraints (Gauss's
laws), lattice gauge theories have an intricate Hilbert-space structure. This
structure complicates the definition of thermodynamic properties of systems
coupled to reservoirs during equilibrium and nonequilibrium processes. We show
how to define thermodynamic quantities such as work and heat using
strong-coupling thermodynamics, a framework that has recently burgeoned within
the field of quantum thermodynamics. Our definitions suit instantaneous
quenches, simple nonequilibrium processes undertaken in quantum simulators. To
illustrate our framework, we compute the work and heat exchanged during a
quench in a $Z_2$ lattice gauge theory coupled to matter in 1+1 dimensions. The
thermodynamic quantities, as functions of the quench parameter, evidence an
expected phase transition. For general thermal states, we derive a simple
relation between a quantum many-body system's entanglement Hamiltonian,
measurable with quantum-information-processing tools, and the Hamiltonian of
mean force, used to define strong-coupling thermodynamic quantities.
