  We investigate how bounds of resonance counting functions for Schottky
surfaces behave under transitions to covering surfaces of finite degree. We
consider the classical resonance counting function asking for the number of
resonances in large (and growing) disks centered at the origin of $\mathbb{C}$,
as well as the (fractal) resonance counting function asking for the number of
resonances in boxes near the axis of the critical exponent. For the former
counting function we provide a transfer-operator-based proof that bounding
constants can be chosen such that the transformation behavior under transition
to covers is as for the Weyl law in the case of surfaces of finite area. For
the latter counting function we deduce a bound in terms of the covering degree
and the minimal length of a periodic geodesic on the covering surface. This
yields an improved fractal Weyl upper bound. In the setting of Schottky
surfaces, these estimates refine previous results due to Guillop\'{e}--Zworski
and Guillop\'{e}--Lin--Zworski. When applied to principal congruence covers,
these results yield new estimates for the resonance counting functions in the
level aspect, which have recently been investigated by Jakobson--Naud. The
techniques used in this article are based on the thermodynamic formalism for
$L$-functions (twisted Selberg zeta functions), and twisted transfer operators.
