Recent advances in audio large language models (ALLMs) have made high-quality synthetic audio widely accessible, increasing the risk of malicious audio deepfakes across speech, environmental sounds, singing voice, and music. Real-world audio deepfake detection (ADD) therefore requires all-type detectors that generalize across heterogeneous audio and provide interpretable decisions. Given the strong multi-task generalization ability of ALLMs, we first investigate their performance on all-type ADD under both supervised fine-tuning (SFT) and reinforcement fine-tuning (RFT). However, SFT using only binary real/fake labels tends to reduce the model to a black-box classifier, sacrificing interpretability. Meanwhile, vanilla RFT under sparse supervision is prone to reward hacking and can produce hallucinated, ungrounded rationales. To address this, we propose an automatic annotation and polishing pipeline that constructs Frequency-Time structured chain-of-thought (CoT) rationales, producing ~340K cold-start demonstrations. Building on CoT data, we propose Frequency Time-Group Relative Policy Optimization (FT-GRPO), a two-stage training paradigm that cold-starts ALLMs with SFT and then applies GRPO under rule-based frequency-time constraints. Experiments demonstrate that FT-GRPO achieves state-of-the-art performance on all-type ADD while producing interpretable, FT-grounded rationales. The data and code are available online.