  The notion of a nonequilibrium heat capacity is important for bio-energetics
and for calorimetry of active materials more generally. It centers around the
notion of excess heat or excess work dissipated during a quasistatic relaxation
between different nonequilibrium conditions. We give exact results for active
random walks moving in an energy landscape on a graph, based on calculations
employing the matrix-tree and matrix-forest theorems. That graphical method
applies to any Markov jump process under the physical condition of local
detailed balance, and is not restricted to the examples given in this paper.
