Hot pursuit refers to the situation when a law enforcement officer can arrest a person without the usual arrest warrant. If a felony has just occurred then an officer can arrest or search for a fleeing suspect to prevent the suspect from escaping.
The following of another in an attempt to overtake and capture: chase, pursuit.
The immediate and continuous pursuit by police officers of a fleeing suspect whose possible escape justifies the failure of the officers to obtain a warrant before making an entry, search, seizure, or arrest.
n. when a law enforcement officer is so close behind the alleged criminal that he/she may continue the chase into another jurisdiction without stopping or seeking a warrant for an arrest in the other county or state. It is equivalent to fresh pursuit.
Immediate pursuit (also known as fresh or hot pursuit) is a common law principle describing rules of engagement to enter into combat with or apprehend and forcibly detain another, invade one's privacy, etc. Immediate pursuit enables, for example, a citizen to arrest a person committing some offence, without incurring legal liability. It also justifies active/aggressive countermeasures to theft or other intrusion.