Hysteresis and eddy losses depend on frequency

Both losses grow with frequency because a higher frequency means the core is magnetised and demagnetised more times each second and the induced emf in the core is larger.

Hysteresis loss

The energy lost in one cycle per unit volume is the area $w_h$ of the $B$–$H$ loop. It depends on the material and $B_{max}$ only. The number of cycles per second is $f$, so

$$P_h = w_h V f = \eta B_{max}^{1.6} f V$$

The loss is directly proportional to $f$. At twice the frequency, the loop is traversed twice as often and the loss doubles (for the same $B_{max}$).

Eddy current loss

The flux in the core is $\phi = \phi_m \sin\omega t$. The emf induced in a lamination is proportional to the rate of change of flux:

$$e \propto \frac{d\phi}{dt} = \omega\phi_m\cos\omega t \Rightarrow E_{eddy} \propto f B_{max}$$

The eddy current is $I_e = E_{eddy}/R_{core}$, and the loss is $I_e^2 R$. So the loss varies as the square of the induced emf:

$$P_e \propto E_{eddy}^2 \propto f^{2} B_{max}^{2}$$

Eddy loss is therefore proportional to $f^2$. At twice the frequency the loss is four times larger.

Note: in a transformer on a fixed supply, $B_{max} \propto V/f$. Then $P_e$ is almost independent of $f$, while $P_h \propto f^{-0.6}$. The statement is therefore true for a given $B_{max}$.