As the first great materialist philosopher, Thomas Hobbes encouraged empiricism and political science. He challenged religious authority and favored an authoritarian state. His political philosophy was based on science, empiricism and secularism. One influence that is seen in the United States today is this idea of church and state.
He also espoused self-preservation via unity in submission to a strong centralized state, likely governed by a monarch. Strong central government is more characteristic of democratic philosophy. Hobbes definitely supported a strong central government with a monarch and thus, must be considered to endorse a kind of liberal dictatorship. These ideas did not influence the United States to the same extent as the separation of church and state. However, you could argue that early American government was a revision of Hobbes' idea. America is decidedly more individualistic with limited central rule. The United States was founded on the concept of free but united states.
Although Hobbes’ ideas of absolute rule are obsolete in terms of individual freedom in American politics, his general concept of unity and equality in Leviathan could be partially influential. The concept of equality in America today is more individualistic. That is, we each should enjoy the privileges, be subject to the same laws and be free from oppression from the state. Under Hobbes’ philosophy, equality was based on unity of individuals in their interdependence on one another and their collective dependence on the absolute authority of the state.
The thinking of Thomas Hobbes is most influential in the field of International Relations. In most other fields of political science, the Hobbesian "state of nature" is so far removed from our present reality that Hobbes's ideas no longer apply. Similarly, his idea that all sovereignty must be handed over to Leviathan is not at all in favor in modern America.
However, many international relations theorists (particularly those in the "realist" camp) find Hobbes's description of the state of nature to be very applicable to the international political order. They argue that states exist in something of a state of nature where there is no force (outside of other, more powerful states) that compel states to act or constrain them from acting. Thus, states are in a situation where might makes right and each state is always in pursuit of power. This matches very closely with Hobbes's view of the world.