地 pronunciation variation
Very interesting question.
Like what @AwQiruiGuo has said, 地 should be pronounced as [de] (in speech, that is). And indeed the mentioned singer does not speak perfect Mandarin.
However, this questions has more complicated background and should not be simply explained as the personal accent of the singer. Because in the case of songs, pronounciation variation kicks in.
Chinese Mandarin is a tonal language. But when the lyrics were sung, rhythm and beats can alter the tones. I'll take the lyric in the question as an example: Dictionary says 那 should be pronouced in 4th tone, but in this song, 那 is in 1st tone. Just listen to this song, and imagine what it would sound like if 那 is sung in 4th tone.
In lyric, when a charater is on the downbeat, it is accented (stressed). If we mark all the downbeats, the lyric in the question will look like this:

冷得[连]隐[藏]的[遗]憾都[那]么[地]明[显]
The characters in square brackets are on downbeats.

In Chinese phonology there is a concept called 平仄, which deals with the arrangement of characters in poems(诗) and lyrics(词) according to their tones. 词 is an art form with strict metrical foot which is comparable to English poetry. You have to write each line of lyrics with specific number of syllables in specific stressed/unstressed order. The guideline of how you arrange the stressed/unstressed syllables was of course the original rhythm and beats of ancient songs, which unfortunately were most - if not all - lost in the history.
Ancient Chinese pronounciation was different from modern Chinese but you can roughly interprete 平 as stressed (1st and 2nd tones in modern Mandarin) and 仄 as unstressed (3rd, 4th and 5th tones in modern Mandarin). Although pop music does not follow ancient meters any more, rhythms and beats still impact pronounciation because it's only nature that people pronounce stressed syllables on downbeats.
With this in mind, now take a look again at the downbeat characters in the mentioned line of lyrics:

冷得[连]隐[藏]的[遗]憾都[那]么[地]明[显]
连: 2nd tone, stressed. In this song, it sounds like 1st tone, still stressed.
藏: 2nd tone, stressed. In this song, it sounds like 1st tone, still stressed.
遗: 2nd tone, stressed. In this song, it sounds like 1st tone, still stressed.
那: 4th tone, unstressed. But in this song, it sounds like 1st tone, stressed.
显: 3rd tone, unstressed. But in this song, it sounds like 1st tone, stressed.
now back to 地 in question: shoule be 5th tone, unstressed. But in this song, it sounds like 1st tone, and is altered from [de] to [di].

Because [de] was 5th tone which is a very light one, if simply pronouced as [de] in 1st tone it just doesn't feel right. So traditionally when you want to stress [de5] you make it [di1]. And not only 地, but even 的 also follows this rule. I think this usage traces back to Chinese Opera. For example in The Legend of the Red Lantern (红灯记) there are some famous lines of lyrics：

穷人的孩子早当家
都有一颗红亮的心

where 的 is pronounced as [di], even if it's not on downbeat.