I think it's because they would use цей for a particular university. But my Ukrainian wife says "I am looking for the university" would be a perfectly good translation for "Я шукаю університет"
Unless you are looking for a university to enrol to. Depends on the context.
Isn't it a rule in English that u use an 'a' before consonants and 'an' for non-consonant (sorry, I do not know the word for it, I am not a native speaker)
As a native speaker, I intuited that the choice of a or an depends on the sound of the following letter not whether it is a vowel (the word you were looking for) or not.
A google search confirmed that my intuition is considered correct.
But my intuition gets me in trouble with "h" ... I say "an historical ...," but others say "a historical." And Google says there are differing opinions on that one.
So, with regard to the word ‘university’, the initial sound is actually the consonant sound /j/, so we say ‘a university’. If we look at your original question, the word ‘LP’ begins with a vowel sound, /e/. That is why we say ‘an LP’. Source: bbc.co.uk http://www.bbc.co.uk/worldservice/learningenglish/language/askaboutenglish/2010/06/100622_aae_indefinite_articles_pron.shtml