  Recent observations suggest that there are violations of the isotropy of the
universe at large scales, an important part of the cosmological principle. In
this paper, we use the Cosmic Microwave Background (CMB) data to search for
spatial variations of the cosmological parameters in the $\Lambda\mathrm{CDM}$
model. We fit the Planck temperature angular power spectrum
$\mathcal{C}^{TT}_\ell$ for 48 different half-skies, centering on 48 different
directions, to search for directional dependences of the standard cosmological
parameters. There are $3(2)\sigma$-level directional variations in
$\Omega_bh^2$, $\Omega_ch^2$, $n_s$, $100\theta_\mathrm{MC}$, and $H_0$ $(\tau$
and $\ln(10^{10}A_s))$. Furthermore, the directional distributions of the
parameters follow a dipole form to good approximation. The Bayes factor between
the isotropic and anisotropic hypotheses is $0.0041$, strongly disfavouring the
former. The best-fit dipole axes for $\Omega_bh^2$, $\Omega_ch^2$, $n_s$,
$100\theta_\mathrm{MC}$, and $A_s e^{-2\tau}$ all generally align with the mean
direction of $\boldsymbol{V} \equiv (b = -5.6^{+17.0{\circ}}_{-17.4}, l =
48.8^{+14.3{\circ}}_{-14.4})$, which is roughly perpendicular to the dipole of
the variation in fine structure constant, and is about $45^{\circ}$ to the
directions of the CMB kinematic dipole, CMB parity asymmetry, and polarization
of QSOs. Our results suggest either significant violation of the cosmological
principle, or previously unknown systematic errors in the standard CMB
analysis.
