Is there a lower bound on the running time for solving 3-SAT if P = NP. For instance, is it known that 3-SAT can't be solved in linear time? What about quadratic?
These are some of the best time and space bounds known. In this area the research has gone to the direction of giving bounds in both time and space. My understanding is that superlinear time bounds without a space constraint are not known.
Time-Space Lower Bounds for Satisﬁability Lance Fortnow, Richard Lipton, Dieter van Melkebeek
We establish the ﬁrst polynomial time-space lower bounds for satisﬁability on general models of computation. We show that for any constant $c$ less than the golden ratio there exists a positive constant $d$ such that no deterministic random-access Turing machine can solve satisﬁability in time $n^c$ and space $n^d$, where $d$ approaches 1 when $c$ does.