Nowadays, GPU accelerators are commonly used to speed up general-purpose computing tasks on a variety of hardware. However, due to the diversity of GPU architectures and processed data, optimization of codes for a particular type of hardware and specific data characteristics can be extremely challenging. The autotuning of performance-relevant source-code parameters allows for automatic optimization of applications and keeps their performance portable. Although the autotuning process typically results in code speed-up, searching the tuning space can bring unacceptable overhead if (i) the tuning space is vast and full of poorly-performing implementations, or (ii) the autotuning process has to be repeated frequently because of changes in processed data or migration to different hardware. In this paper, we introduce a novel method for searching generic tuning spaces. The tuning spaces can contain tuning parameters changing any user-defined property of the source code. The method takes advantage of collecting hardware performance counters (also known as profiling counters) during empirical tuning. Those counters are used to navigate the searching process towards faster implementations. The method requires the tuning space to be sampled on any GPU. It builds a problem-specific model, which can be used during autotuning on various, even previously unseen inputs or GPUs. Using a set of five benchmarks, we experimentally demonstrate that our method can speed up autotuning when an application needs to be ported to different hardware or when it needs to process data with different characteristics. We also compared our method to state of the art and show that our method is superior in terms of the number of searching steps and typically outperforms other searches in terms of convergence time.