Emerging non-volatile memories (NVMs) have favorable properties, such as low leakage and high density, and have attracted a lot of attention in recent years. Among them, spin-transfer torque magnetoresistive random access memory (STT-MRAM) with SRAM-comparable read speed is a good candidate to build large last-level caches (LLCs). However, STT-MRAM suffers from long write latency and high write energy. To mitigate the impact of asymmetric read/write energy and latency, hybrid cache designs have been proposed to combine the merits of STT-MRAM and SRAM. In such a hybrid SRAM/STT-MRAM LLC, intelligent block placement and migration policies are needed to improve the energy efficiency. Prior studies map write-intensive blocks to SRAM and keep read-intensive blocks in STT-MRAM for reducing the energy consumption of hybrid LLCs. The write-intensive/read-intensive blocks are usually captured by sampling the address (PC) of memory access instructions or adding simple access counters in each cache line. Nevertheless, these prior approaches cannot fully capture the energy-harmful access behavior in STT-MRAM, especially the writes caused by repetitive data transfer between the LLC and upper-level caches. In this paper, we find that conflict misses in L2 often generate thrashing blocks which move back and forth between L2 and LLC. If dirty thrashing blocks that incur extensive writes are placed in STT-MRAM, energy consumption would excessively increase, especially when running memory-bound workloads. Thus, we propose a thrashing aware placement and migration policy (TAP) to tackle the challenge. TAP places dirty thrashing blocks into SRAM and migrates clean thrashing blocks from SRAM to STT-MRAM. Evaluation results show that TAP can provide significant energy savings with minimal performance loss.