We present measurements of direct photon pair production cross sections using 8.5 fb$^{-1}$ of data collected with the D0 detector at the Fermilab Tevatron $p \bar p$ collider. The results are presented as differential distributions of the photon pair invariant mass $d\sigma/dM_{\gamma \gamma}$, pair transverse momentum $d \sigma /dp^{\gamma \gamma}_T$, azimuthal angle between the photons $d\sigma/d\Delta \phi_{\gamma \gamma}$, and polar scattering angle in the Collins-Soper frame $d\sigma /d|\cos \theta^*|$. Measurements are performed for isolated photons with transverse momenta $p^{\gamma}_T>18 ~(17)$ GeV for the leading (next-to-leading) photon in $p_T$, pseudorapidities $|\eta^{\gamma}|<0.9$, and a separation in $\eta-\phi$ space $\Delta\mathcal R_{\gamma\gamma} > 0.4$. We present comparisons with the predictions from Monte Carlo event generators {\sc diphox} and {\sc resbos} implementing QCD calculations at next-to-leading order, $2\gamma${\sc nnlo} at next-to-next-to-leading order, and {\sc sherpa} using matrix elements with higher-order real emissions matched to parton shower.