Measurement of the CMB temperature power spectrum and constraints on cosmology from the SPT-3G 2018 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mrow><mml:mi>T</mml:mi><mml:mi>T</mml:mi></mml:mrow></mml:mrow></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>T</mml:mi><mml:mi>E</mml:mi></mml:mrow></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>E</mml:mi><mml:mi>E</mml:mi></mml:mrow></mml:math> dataset

We present a sample-variance-limited measurement of the temperature power spectrum ($TT$) of the cosmic microwave background using observations of a $\ensuremath{\sim}1500\text{ }\text{ }{\mathrm{deg}}^{2}$ field made by the SPT-3G in 2018. We report multifrequency power spectrum measurements at 95, 150, and 220 GHz covering the angular multipole range $750\ensuremath{\le}\ensuremath{\ell}&lt;3000$. We combine this $TT$ measurement with the published polarization power spectrum measurements from the 2018 observing season and update their associated covariance matrix to complete the SPT-3G 2018 $TT/TE/EE$ dataset. This is the first analysis to present cosmological constraints from SPT $TT$, $TE$, and $EE$ power spectrum measurements jointly. We blind the cosmological results and subject the dataset to a series of consistency tests at the power spectrum and parameter level. We find excellent agreement between frequencies and spectrum types and our results are robust to the modeling of astrophysical foregrounds. We report results for $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ and a series of extensions, drawing on the following parameters: the amplitude of the gravitational lensing effect on primary power spectra ${A}_{\mathrm{L}}$, the effective number of neutrino species ${N}_{\mathrm{eff}}$, the primordial helium abundance ${Y}_{\mathrm{P}}$, and the baryon clumping factor due to primordial magnetic fields $b$. We find that the SPT-3G 2018 $TT/TE/EE$ data are well fit by $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ with a probability to exceed of 15%. For $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$, we constrain the expansion rate today to ${H}_{0}=68.3\ifmmode\pm\else\textpm\fi{}1.5\text{ }\text{ }\mathrm{km}\text{ }{\mathrm{s}}^{\ensuremath{-}1}\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$ and the combined structure growth parameter to ${S}_{8}=0.797\ifmmode\pm\else\textpm\fi{}0.042$. The SPT-based results are effectively independent of Planck, and the cosmological parameter constraints from either dataset are within $&lt;1\ensuremath{\sigma}$ of each other. The addition of temperature data to the SPT-3G $TE/EE$ power spectra improves constraints by 8--27% for each of the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ cosmological parameters. When additionally fitting ${A}_{\mathrm{L}}$, ${N}_{\mathrm{eff}}$, or ${N}_{\mathrm{eff}}+{Y}_{\mathrm{P}}$, the posteriors of these parameters tighten by 5--24%. In the case of primordial magnetic fields, complete $TT/TE/EE$ power spectrum measurements are necessary to break the degeneracy between $b$ and ${n}_{s}$, the spectral index of primordial density perturbations. We report a 95% confidence upper limit from SPT-3G data of $b&lt;1.0$. The cosmological constraints in this work are the tightest from SPT primary power spectrum measurements to date and the analysis forms a new framework for future SPT analyses.