Ironically, one is indeed likely to find signs of life at the bottom of any coffee cup, cold or otherwise.

Three distinct paths by which this is so are immediate.

Firstly, there is the existence of an undeniable product of intelligent life, this being the cup of cold coffee conceptually. If a visitor to an alien asteroid were ever to discover a sculpture hewn in dead and frozen rock of "a cup of cold coffee" -- even though the edifice had itself never lived, nor harbored life -- it would yet be apparent by the presence of the subject, a smoothly crafted functional structure, that intelligent life had visited there. But here on Earth the existence of at least somewhat intelligent life is a mundanity, and the cup of coffee is no more necessary to prove this phenomenon than is any other piece of detritus.

And secondly, where we earthly find this thing, examination of the cup and the coffee will suffice to demonstrate that both are made from (formerly) living things. The rapid development of molecular DNA isolating technology has, in the past decade, made it possible to take samples of even processed wood and derive some genetic information from it. From this, it might be determined the species of trees from which the paper was milled, possibly the geographic locale of its harvesting, and possibly even whether the trees from which a particular sample were made were commercially grown or even illegally harvested. The same ought to apply for the coffee itself, with molecular fingerprints pointing to the nature of the parent plant for those beans; and the same would go even for added milk and sugar. But the underlying point of all this is that it is now possible to unequivocally demonstrate that the cup, the coffee, and other ingredients were all themselves once part of living things, and so signs of life. Unless the cup is styrofoam, in which event mankind is screwed.

And lastly, putting aside all of the interpolations of human activity, there is no doubt given the tenacity of microbiological life on Earth that by the time that coffee has reached coldness, it is well-populated from top to bottom with some especially hardy, coffee-loving strains of bacteria. For those searching for such signs of life, I would recommend a reasonably powerful microscope. A magnification of 100x will usually do to view the little dudes scooting about on a slide.