  The thermal Sunyaev-Zeldovich effect contains information about the thermal
history of the universe, observable in maps of the Compton $y$ parameter;
however, it does not contain information about the redshift of the sources.
Recent papers have utilized a tomographic approach, cross-correlating the
Compton $y$ map with the locations of galaxies with known redshift, in order to
deproject the signal along the line of sight. In this paper, we test the
validity and accuracy of this tomographic approach to probe the thermal history
of the universe. We use the state-of-the-art cosmological hydrodynamical
simulation, Magneticum, for which the thermal history of the universe is a
known quantity. The key ingredient is the Compton-$y$-weighted halo bias,
$b_y$, computed from the halo model. We find that, at redshifts currently
available, the method reproduces the correct mean thermal pressure (or the
density-weighted mean temperature) to high accuracy, validating and confirming
the results of previous papers. At higher redshifts ($z\gtrsim 2.5$), there is
significant disagreement between $b_y$ from the halo model and the simulation.
