  We investigate the experimental setup proposed in [New J. Phys., 15, 115006
(2013)] for calorimetric measurements of thermodynamic indicators in an open
quantum system. As theoretical model we consider a periodically driven qubit
coupled with a large yet finite electron reservoir, the calorimeter. The
calorimeter is initially at equilibrium with an infinite phonon bath. As time
elapses, the temperature of the calorimeter varies in consequence of energy
exchanges with the qubit and the phonon bath. We show how under weak coupling
assumptions, the evolution of the qubit-calorimeter system can be described by
a generalized quantum jump process including as dynamical variable the
temperature of the calorimeter. We study the jump process by numeric and
analytic methods. Asymptotically with the duration of the drive, the
qubit-calorimeter attains a steady state. In this same limit, we use multiscale
perturbation theory to derive a Fokker--Planck equation governing the
calorimeter temperature distribution. We inquire the properties of the
temperature probability distribution close and at the steady state. In
particular, we predict the behavior of measurable statistical indicators versus
the qubit-calorimeter coupling constant.
