The computational cost of softmax-based attention in transformers limits their applicability to long-context tasks. Adaptive sparsity, of which $α$-entmax attention is an example, offers a flexible data-dependent alternative, but existing implementations are inefficient and do not leverage the sparsity to obtain runtime and memory gains. In this work, we propose AdaSplash, which combines the efficiency of GPU-optimized algorithms with the sparsity benefits of $α$-entmax. We first introduce a hybrid Halley-bisection algorithm, resulting in a 7-fold reduction in the number of iterations needed to compute the $α$-entmax transformation. Then, we implement custom Triton kernels to efficiently handle adaptive sparsity. Experiments with RoBERTa and ModernBERT for text classification and single-vector retrieval, along with GPT-2 for language modeling, show that our method achieves substantial improvements in runtime and memory efficiency compared to existing $α$-entmax implementations. It approaches -- and in some cases surpasses -- the efficiency of highly optimized softmax implementations like FlashAttention-2, enabling long-context training while maintaining strong task performance.