Hyperspectral absorption tomography (HAT) reconstructs the distribution of key gas parameters, including composition, pressure, and temperature, from multi-beam absorbance data with numerous spectral resolution elements. There is a nonlinear relationship between the parameters of interest and the spectral absorption coefficient, which must be incorporated into the tomography algorithm. Nonlinear HAT simultaneously reconstructs the composition and temperature of a gas by minimizing a single nonconvex objective function, which combines the light attenuation and spectroscopy models, using a metaheuristic technique. The time required for this computation depends, strongly, on the assumed heuristics, but the high computational cost limits the problem size and, hence, the obtainable spatial resolution. Conversely, linear HAT reconstructs the absorption coefficient for each measurement wavenumber, individually, exploiting the linear structure of the underlying tomography problem. Local spectra are then post-processed with a spectroscopic model to recover multiple parameters. The linear technique enables accurate reconstructions on a high-resolution grid by way of an established statistical imaging algorithm. Moreover, local spectra can be employed to gauge phenomena such as multi-species broadening and line mixing with a calibrated regression model. We simulate linear and nonlinear HAT and reconstruct experimental absorbance data using the former approach to demonstrate its superior performance. Nonlinear reconstructions required a 100-fold computational effort compared to linear HAT. In our experimental test, we reconstructed the mole fraction, pressure, and temperature of water vapor in a stagnation flow, which represents the first three-parameter laser absorption tomography experiment. Simulated and experimental results in our paper make a comprehensive case for linear HAT compared to the nonlinear method.