Newton's law of cooling is a special case of

According to Stefan's law, emissive power (heat lost per unit time)

$\mathrm{E}=\mathrm{\sigma \epsilon A}\left({\mathrm{T}}^4-{\mathrm{T}}_0^4\right)$, where $\mathrm{T}$ is temperature of the body and ${\mathrm{T}}_0$ that of the surrounding.
$-\mathrm{mc}\frac{\mathrm{dT}}{\mathrm{dt}}=\mathrm{\sigma \epsilon A}\left({\mathrm{T}}^4-{\mathrm{T}}_0^4\right)$

$-\mathrm{mc}\frac{\mathrm{dT}}{\mathrm{dt}}=\mathrm{\sigma \epsilon A}\left[{\left({\mathrm{T}}_0+∆\mathrm{T}\right)}^4-{\mathrm{T}}_0^4\right]$

, where $∆\mathrm{T}$ is excess of temperature of the body compared to its surrounding.

If $∆\mathrm{T}<<{\mathrm{T}}_0$, we can apply Binomial approximation

$-\mathrm{mc}\frac{\mathrm{dT}}{\mathrm{dt}}=\mathrm{\sigma \epsilon A}\left[{\left({\mathrm{T}}_0+∆\mathrm{T}\right)}^4-{\mathrm{T}}_0^4\right]$

$-\mathrm{mc}\frac{\mathrm{dT}}{\mathrm{dt}}=\mathrm{\sigma \epsilon A}\left[{\mathrm{T}}_0^4{\left(1+\frac{∆\mathrm{T}}{{\mathrm{T}}_0}\right)}^4-{\mathrm{T}}_0^4\right]$

$-\mathrm{mc}\frac{\mathrm{dT}}{\mathrm{dt}}=\mathrm{\sigma \epsilon A}\times 4\left(∆\mathrm{T}\right){\mathrm{T}}_0^3$

$\frac{\mathrm{dT}}{\mathrm{dt}}=-\frac{4{\mathrm{\sigma \epsilon AT}}_0^3}{\mathrm{mc}}\left(∆\mathrm{T}\right)\propto \left(∆\mathrm{T}\right)$   This is Newton's law of cooling.