  A notable phenomenon in topological semimetals is the violation of
Kohler$^,$s rule, which dictates that the magnetoresistance $MR$ obeys a
scaling behavior of $MR = f(H/\rho_0$), where $MR = [\rho_H-\rho_0]/\rho_0$ and
$H$ is the magnetic field, with $\rho_H$ and $\rho_0$ being the resistivity at
$H$ and zero field, respectively. Here we report a violation originating from
thermally-induced change in the carrier density. We find that the
magnetoresistance of the Weyl semimetal, TaP, follows an extended Kohler$^,$s
rule $MR = f[H/(n_T\rho_0)]$, with $n_T$ describing the temperature dependence
of the carrier density. We show that $n_T$ is associated with the Fermi level
and the dispersion relation of the semimetal, providing a new way to reveal
information on the electronic bandstructure. We offer a fundamental
understanding of the violation and validity of Kohler$^,$s rule in terms of
different temperature-responses of $n_T$. We apply our extended Kohler$^,$s
rule to BaFe$_2$(As$_{1-x}$P$_x$)$_2$ to settle a long-standing debate on the
scaling behavior of the normal-state magnetoresistance of a superconductor,
namely, $MR$ ~ $tan^2\theta_H$, where $\theta_H$ is the Hall angle. We further
validate the extended Kohler$^,$s rule and demonstrate its generality in a
semiconductor, InSb, where the temperature-dependent carrier density can be
reliably determined both theoretically and experimentally.
