  We present a theoretical investigation of the magnetocrystalline anisotropy
(MA) in $R_2$Fe$_{14}$B ($R$ is a rare-earth element) magnets in consideration
of the non-collinearity effect (NCE) between the $R$ and Fe magnetization
directions. In particular, the temperature dependence of the MA of
Dy$_2$Fe$_{14}$B magnets is detailed in terms of the $n$th-order MA constant
(MAC) $K_n(T)$ at a temperature $T$. The features of this constant are as
follows: $K_1(T)$ has a broad plateau in the low-temperature range and $K_2(T)$
persistently survives in the high-temperature range. The present theory
explains these features in terms of the NCE on the MA by using numerical
calculations for the entire temperature range, and further, by using a
high-temperature expansion. The high-temperature expansion for $K_n(T)$ is
expressed in the form of
$K_n(T)=\kappa_1(T)\left[1+\delta(T)\right][-\delta(T)]^{n-1}$, where
$\kappa_1(T)$ is the part without the NCE and $\delta(T)$ is a correction
factor for the NCE introduced in this study. We also provide a convenient
expression to evaluate $K_n(T)$, which can be determined only by a second-order
crystalline electric field coefficient and an effective exchange field.
