  Stochastic thermodynamics is formulated under the assumption of perfect
knowledge of all thermodynamic parameters. However, in any real-world
experiment, there is non-zero uncertainty about the precise value of
temperatures, chemical potentials, energy spectrum, etc. Here we investigate
how this uncertainty modifies the theorems of stochastic thermodynamics. We
consider two scenarios: in the (called \emph{effective}) scenario we fix the
(unknown, randomly generated) experimental apparatus and then repeatedly
observe (stochastic) trajectories of the system for that fixed apparatus. In
contrast, in a (called \emph{phenomenological}) scenario the (unknown)
apparatus is re-generated for each trajectory. We derive expressions for
thermodynamic quantities in both scenarios. We also discuss the physical
interpretation of effective (scenario) entropy production (EP), derive the
effective mismatch cost, and provide a numerical analysis of the effective
thermodynamics of a quantum dot implementing bit erasure with uncertain
temperature. We then analyze the protocol for moving between two state
distributions that maximize effective work extraction. Next, we investigate the
effective thermodynamic value of information, focusing on the case where there
is a delay between the initialization of the system and the start of the
protocol. Finally, we derive the detailed and integrated fluctuation theorems
(FTs) for the phenomenological EP. In particular, we show how the
phenomenological FTs account for the fact that the longer a trajectory runs,
the more information it provides concerning the precise experimental apparatus,
and therefore the less EP it generates.
