IEEE 802.11p standard is specially developed to define vehicular communications requirements and support cooperative intelligent transport systems. In such environment, reliable channel estimation is considered as a major critical challenge for ensuring the system performance due to the extremely time-varying characteristic of vehicular channels. The channel estimation of IEEE 802.11p is preamble based, which becomes inaccurate in high mobility scenarios. The major challenge is to track the channel variations over the course of packet length while adhering to the standard specifications. The motivation behind this paper is to overcome this issue by proposing a novel deep learning based channel estimation scheme for IEEE 802.11p that optimizes the use of deep neural networks (DNN) to accurately learn the statistics of the spectral temporal averaging (STA) channel estimates and to track their changes over time. Simulation results demonstrate that the proposed channel estimation scheme STA-DNN significantly outperforms classical channel estimators in terms of bit error rate. The proposed STA-DNN architectures also achieve better estimation performance than the recently proposed auto-encoder DNN based channel estimation with at least 55.74% of computational complexity decrease. INDEX TERMS Channel estimation, deep learning, DNN, IEEE 802.11p standard, vehicular channels.