HII region electron temperatures are a critical ingredient in metallicity determinations and recent observations reveal systematic variations in the temperatures measured using different ions. We present electron temperatures ($T_e$) measured using the optical auroral lines ([NII]$\lambda5756$, [OII]$\lambda\lambda7320,7330$, [SII]$\lambda\lambda4069,4076$, [OIII]$\lambda4363$, and [SIII]$\lambda6312$) for a sample of HII regions in seven nearby galaxies. We use observations from the Physics at High Angular resolution in Nearby Galaxies survey (PHANGS) obtained with integral field spectrographs on Keck (Keck Cosmic Web Imager; KCWI) and the Very Large Telescope (Multi-Unit Spectroscopic Explorer; MUSE). We compare the different $T_e$ measurements with HII region and interstellar medium environmental properties such as electron density, ionization parameter, molecular gas velocity dispersion, and stellar association/cluster mass and age obtained from PHANGS. We find that the temperatures from [OII] and [SII] are likely over-estimated due to the presence of electron density inhomogeneities in HII regions. We observe that differences between [NII] and [SIII] temperatures are weakly correlated with stellar association mass and molecular gas velocity dispersion. We measure high [OIII] temperatures in a subset of regions with high molecular gas velocity dispersion and low ionization parameter, which may be explained by the presence of low-velocity shocks. In agreement with previous studies, the $T_{\rm{e}}$--$T_{\rm{e}}$ between [NII] and [SIII] temperatures have the lowest observed scatter and generally follow predictions from photoionization modeling, which suggests that these tracers reflect HII region temperatures across the various ionization zones better than [OII], [SII], and [OIII].
Comment: Accepted to ApJ, 66 pages, 36 figures