  We consider curvature corrections to static, axisymmetric Dirac-Nambu-Goto
membranes embedded into a spherically symmetric black hole spacetime with
arbitrary number of dimensions. Since the next to leading order corrections in
the effective brane action are quadratic in the brane thickness l, we adopt a
linear perturbation approach in l^2. The perturbations are general in the sense
that they are not restricted to the Rindler zone nor to the near-critical
solutions of the unperturbed system. As a result, an unexpected asymmetry in
the perturbed system is found. In configurations, where the brane does not
cross the black hole horizon, the perturbative approach does not lead to
regular solutions if the number of the brane's spacetime dimensions D>3. This
condition, however, does not hold for the horizon crossing solutions.
Consequently we argue that the presented perturbative approach breaks down for
subcritical type solutions near the axis of the system for D>3. Nevertheless,
we can discuss topology-changing phase transitions in cases when D=2 or 3, i.e.
when the brane is a 1-dimensional string or a 2-dimensional sheet,
respectively. For the general case, a different, non-perturbative approach
should be sought. Based on the energy properties of those branes that are
quasi-statically evolved from the equatorial configuration, we illustrate the
results of the phase transition in the case of a D=3 brane. It is found that
small thickness perturbations do not modify the order of the transition, i.e.
it remains first order just as in the case of vanishing thickness.
