  We develop a numerical method to calculate the Bethe logarithm for resonant
states. We use the Complex Coordinate Rotation (CCR) formalism to describe
resonances as time-independent Schr\"odinger solutions. To get a proper
expression for the Bethe logarithm we apply the generalization of the second
order perturbation theory to an isolated CCR eigenstate. Using the developed
method we perform a systematic calculation of the Bethe logarithm for
metastable states in the antiprotonic helium He$^+\bar{p}\,$ atoms with
precision of 7--8 significant digits. We also recalculate the nonrelativistic
energies with improved precision using CODATA10 recommended values of masses.
Along with a complete set of corrections of $m\alpha^7$ order and the leading
contributions of $m\alpha^8$ order that has allowed us to get new theoretical
values for ro-vibrational transition frequencies for the He$^+\bar{p}$ atoms
with uncertainty of 0.1--0.3 MHz.
